99 solved questions on Application of Derivatives, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
If the point P on the curve, 4x2 + 5y2 = 20 is farthest from the point Q(0, -4), then PQ2 is equal to:
View solution →Let $f:[2,4] \rightarrow \mathbb{R}$ be a differentiable function such that $$\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)+f(x) \geq 1, x \in[2,4]$$ with…
View solution →If 'R' is the least value of 'a' such that the function f(x) = x2 + ax + 1 is increasing on [1, 2] and 'S' is the greatest value of 'a' such that the function f(x) = x2 + ax + 1…
View solution →If the local maximum value of the function $$f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right)$$ , is $\frac{k}{e}$, then…
View solution →Consider the function $f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}$ defined by $f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$. Consider the statements (I) The curve $y=f(x)$…
View solution →The function $f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}$
View solution →Let $f(x)=3 \sqrt{x-2}+\sqrt{4-x}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha^2+2 \beta^2$ is…
View solution →Let the set of all values of $p$, for which $f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7$ does not have any critical point, be the interval $(a,…
View solution →The slope of tangent at any point (x, y) on a curve $y=y(x)$ is ${{{x^2} + {y^2}} \over {2xy}},x 0$. If $y(2) = 0$, then a value of $y(8)$ is :
View solution →The length of the perpendicular from the origin, on the normal to the curve, x2 + 2xy – 3y2 = 0 at the point (2,2) is
View solution →Let f : [$-$1, 1] $\to$ R be defined as f(x) = ax2 + bx + c for all x$\in$[$-$1, 1], where a, b, c$\in$R such that f($-$1) = 2, f'($-$1) = 1 for x$\in$($-$1, 1) the maximum value…
View solution →Let the function $f(x)=2 x^{2}-\log _{\mathrm{e}} x, x0$, be decreasing in $(0, \mathrm{a})$ and increasing in $(\mathrm{a}, 4)$. A tangent to the parabola $y^{2}=4 a x$ at a…
View solution →Let $M$ and $N$ be the number of points on the curve $y^{5}-9 x y+2 x=0$, where the tangents to the curve are parallel to $x$-axis and $y$-axis, respectively. Then the value of…
View solution →Let $g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)$ and $f^{\prime \prime}(x)0$ for all $x \in(0,3)$. If $g$ is decreasing in $(0, \alpha)$ and increasing in $(\alpha, 3)$, then $8…
View solution →If the tangent to the curve y = x + sin y at a point (a, b) is parallel to the line joining $\left( {0,{3 \over 2}} \right)$ and $\left( {{1 \over 2},2} \right)$, then :
View solution →The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7…
View solution →If the surface area of a cube is increasing at a rate of 3.6 cm2/sec, retaining its shape; then the rate of change of its volume (in cm3/sec), when the length of a side of the…
View solution →Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their…
View solution →Let $f(x) = |(x - 1)({x^2} - 2x - 3)| + x - 3,\,x \in R$. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m +…
View solution →If the absolute maximum value of the function $$f(x)=\left(x^{2}-2 x+7\right) \mathrm{e}^{\left(4 x^{3}-12 x^{2}-180 x+31\right)}$$ in the interval $[-3,0]$ is $f(\alpha)$, then :
View solution →Let $f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10$, $x \in [ - 1,1]$. If [a, b] is the range of the function f, then 4a $-$ b is equal to :
View solution →The function $f(x)=x \mathrm{e}^{x(1-x)}, x \in \mathbb{R}$, is :
View solution →Let f : (–1, $\infty$) $\to$ R be defined by f(0) = 1 and f(x) = ${1 \over x}{\log _e}\left( {1 + x} \right)$, x $\ne$ 0. Then the function f :
View solution →For the function $f(x)=(\cos x)-x+1, x \in \mathbb{R}$, between the following two statements (S1) $f(x)=0$ for only one value of $x$ in $[0, \pi]$. (S2) $f(x)$ is decreasing in…
View solution →Let $\lambda$$^ *$ be the largest value of $\lambda$ for which the function ${f_\lambda }(x) = 4\lambda {x^3} - 36\lambda {x^2} + 36x + 48$ is increasing for all x $\in$ R. Then…
View solution →The sum of the maximum and minimum values of the function $f(x)=|5 x-7|+\left[x^{2}+2 x\right]$ in the interval $\left[\frac{5}{4}, 2\right]$, where $[t]$ is the greatest integer…
View solution →The interval in which the function $f(x)=x^x, x0$, is strictly increasing is
View solution →Let ƒ(x) = xcos–1(–sin|x|), $x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]$, then which of the following is true?
View solution →A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness the melts at a rate of 50 cm3/min. When the thickness of ice is 5 cm, then the rate (in…
View solution →Let the function $f(x) = 2{x^3} + (2p - 7){x^2} + 3(2p - 9)x - 6$ have a maxima for some value of $x 0$. Then, the set of all values of p is
View solution →Let the quadratic curve passing through the point $(-1,0)$ and touching the line $y=x$ at $(1,1)$ be $y=f(x)$. Then the $x$-intercept of the normal to the curve at the point…
View solution →Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, $-$3) and (4, $-$2$\sqrt 2$), and given that a $-$ 2$\sqrt 2$…
View solution →Let S be the set of all the natural numbers, for which the line ${x \over a} + {y \over b} = 2$ is a tangent to the curve ${\left( {{x \over a}} \right)^n} + {\left( {{y \over b}}…
View solution →Let for a differentiable function $f:(0, \infty) \rightarrow \mathbf{R}, f(x)-f(y) \geqslant \log _{\mathrm{e}}\left(\frac{x}{y}\right)+x-y, \forall x, y \in(0, \infty)$. Then…
View solution →Let $f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x \in \mathbf{R}$. Then $\frac{g(7)}{g^{\prime}(7)}$ is equal to :
View solution →If the equation of the normal to the curve $y = {{x - a} \over {(x + b)(x - 2)}}$ at the point (1, $-$3) is $x - 4y = 13$, then the value of $a + b$ is equal to ___________.
View solution →The curve $y(x)=a x^{3}+b x^{2}+c x+5$ touches the $x$-axis at the point $\mathrm{P}(-2,0)$ and cuts the $y$-axis at the point $Q$, where $y^{\prime}$ is equal to 3 . Then the…
View solution →If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to :
View solution →$$\max _\limits{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=$$
View solution →If the curves, ${{{x^2}} \over a} + {{{y^2}} \over b} = 1$ and ${{{x^2}} \over c} + {{{y^2}} \over d} = 1$ intersect each other at an angle of 90$^\circ$, then which of the…
View solution →Let slope of the tangent line to a curve at any point P(x, y) be given by ${{x{y^2} + y} \over x}$. If the curve intersects the line x + 2y = 4 at x = $-$2, then the value of y,…
View solution →If the function $f(x)=2 x^3-9 \mathrm{ax}^2+12 \mathrm{a}^2 x+1, \mathrm{a} 0$ has a local maximum at $x=\alpha$ and a local minimum at $x=\alpha^2$, then $\alpha$ and $\alpha^2$…
View solution →Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$ be $\frac{m}{n}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$.…
View solution →Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) $\ge$ 1 and f''(x) $\ge$ 4, for all x $\in$ (1, 6), then :
View solution →Let a be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ___________.
View solution →Which of the following points lies on the tangent to the curve x4ey + 2$\sqrt {y + 1}$ = 3 at the point (1, 0)?
View solution →The slope of normal at any point (x, y), x 0, y 0 on the curve y = y(x) is given by ${{{x^2}} \over {xy - {x^2}{y^2} - 1}}$. If the curve passes through the point (1, 1), then e .…
View solution →If c is a point at which Rolle's theorem holds for the function, f(x) = ${\log _e}\left( {{{{x^2} + \alpha } \over {7x}}} \right)$ in the interval [3, 4], where a $\in$ R, then…
View solution →Let f(x) be a cubic polynomial with f(1) = $-$10, f($-$1) = 6, and has a local minima at x = 1, and f'(x) has a local minima at x = $-$1. Then f(3) is equal to ____________.
View solution →Let the function, ƒ:[-7, 0]$\to$R be continuous on [-7,0] and differentiable on (-7, 0). If ƒ(-7) = - 3 and ƒ'(x) $\le$ 2, for all x $\in$ (-7,0), then for all such functions…
View solution →Let f : ℝ $\to$ ℝ be a polynomial function of degree four having extreme values at x = 4 and x = 5. If $ \lim\limits_{x \to 0} \frac{f(x)}{x^2} = 5 $, then f(2) is equal to :
View solution →Let $f:R \to R$ be defined as$$f(x) = \left\{ {\matrix{ { - 55x,} & {if\,x < - 5} \cr {2{x^3} - 3{x^2} - 120x,} & {if\, - 5 \le x \le 4} \cr {2{x^3} - 3{x^2} - 36x - 336,} &…
View solution →Let $x=2$ be a local minima of the function $f(x)=2x^4-18x^2+8x+12,x\in(-4,4)$. If M is local maximum value of the function $f$ in ($-4,4)$, then M =
View solution →Let ƒ(x) be a polynomial of degree 3 such that ƒ(–1) = 10, ƒ(1) = –6, ƒ(x) has a critical point at x = –1 and ƒ'(x) has a critical point at x = 1. Then ƒ(x) has a…
View solution →If xy4 attains maximum value at the point (x, y) on the line passing through the points (50 + $\alpha$, 0) and (0, 50 + $\alpha$), $\alpha$ 0, then (x, y) also lies on the line :
View solution →Let f be any function defined on R and let it satisfy the condition : $|f(x) - f(y)|\, \le \,|{(x - y)^2}|,\forall (x,y) \in R$If f(0) = 1, then :
View solution →The number of critical points of the function $f(x)=(x-2)^{2 / 3}(2 x+1)$ is
View solution →If the tangent to the curve, y = f (x) = xloge x, (x > 0) at a point (c, f(c)) is parallel to the line-segment joining the points (1, 0) and (e, e), then c is equal to :
View solution →The function f(x) = ${{4{x^3} - 3{x^2}} \over 6} - 2\sin x + \left( {2x - 1} \right)\cos x$ :
View solution →The value of c in the Lagrange's mean value theorem for the function ƒ(x) = x3 - 4x2 + 8x + 11, when x $\in$ [0, 1] is:
View solution →Let $\mathrm{g}(x)=f(x)+f(1-x)$ and $f^{\prime \prime}(x) 0, x \in(0,1)$. If $\mathrm{g}$ is decreasing in the interval $(0, a)$ and increasing in the interval $(\alpha, 1)$, then…
View solution →The sum of absolute maximum and absolute minimum values of the function $f(x) = |2{x^2} + 3x - 2| + \sin x\cos x$ in the interval [0, 1] is :
View solution →Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x = $-$1 and x = 1. If $\mathop {\lim }\limits_{x \to 0} {{f(x)} \over…
View solution →Let a curve $y=f(x), x \in(0, \infty)$ pass through the points $P\left(1, \frac{3}{2}\right)$ and $Q\left(a, \frac{1}{2}\right)$. If the tangent at any point $R(b, f(b))$ to the…
View solution →If the curve y = ax2 + bx + c, x$\in$R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are :
View solution →The sum of the absolute maximum and minimum values of the function $f(x)=\left|x^{2}-5 x+6\right|-3 x+2$ in the interval $[-1,3]$ is equal to :
View solution →The maximum slope of the curve $y = {1 \over 2}{x^4} - 5{x^3} + 18{x^2} - 19x$ occurs at the point :
View solution →Consider the function f : R $\to$ R defined by $$f(x) = \left\{ \matrix{ \left( {2 - \sin \left( {{1 \over x}} \right)} \right)|x|,x \ne 0 \hfill \cr 0,\,\,x = 0 \hfill \cr}…
View solution →The minimum value of $\alpha$ for which the equation ${4 \over {\sin x}} + {1 \over {1 - \sin x}} = \alpha$ has at least one solution in $\left( {0,{\pi \over 2}} \right)$ is…
View solution →The set of all real values of $\lambda$ for which the function $$f(x) = \left( {1 - {{\cos }^2}x} \right)\left( {\lambda + \sin x} \right),x \in \left( { - {\pi \over 2},{\pi…
View solution →If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$ and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$ have a common extreme point, then $a+2 b+7$ is equal to :
View solution →If the tangent to the curve $y=x^{3}-x^{2}+x$ at the point $(a, b)$ is also tangent to the curve $y = 5{x^2} + 2x - 25$ at the point (2, $-$1), then $|2a + 9b|$ is equal to…
View solution →Let $\lambda x - 2y = \mu$ be a tangent to the hyperbola ${a^2}{x^2} - {y^2} = {b^2}$. Then ${\left( {{\lambda \over a}} \right)^2} - {\left( {{\mu \over b}} \right)^2}$ is equal…
View solution →The maximum value of $$f(x) = \left| {\matrix{ {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\cos 2x} \cr {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos…
View solution →Let 'a' be a real number such that the function f(x) = ax2 + 6x $-$ 15, x $\in$ R is increasing in $\left( { - \infty ,{3 \over 4}} \right)$ and decreasing in $\left( {{3 \over…
View solution →A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is…
View solution →The number of points on the curve $y=54 x^{5}-135 x^{4}-70 x^{3}+180 x^{2}+210 x$ at which the normal lines are parallel to $x+90 y+2=0$ is :
View solution →The sum of all the local minimum values of the twice differentiable function f : R $\to$ R defined by $f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x + f''(1)$ is :
View solution →The position of a moving car at time t is given by f(t) = at2 + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average speed of the car over the time…
View solution →If the angle made by the tangent at the point (x0, y0) on the curve $x = 12(t + \sin t\cos t)$, $y = 12{(1 + \sin t)^2}$, $0 0 is equal to:
View solution →If the lines x + y = a and x – y = b touch the curve y = x2 – 3x + 2 at the points where the curve intersects the x-axis, then ${a \over b}$ is equal to _______.
View solution →A wire of length $20 \mathrm{~m}$ is to be cut into two pieces. A piece of length $l_{1}$ is bent to make a square of area $A_{1}$ and the other piece of length $l_{2}$ is made…
View solution →For the function $$f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],$$ consider the following two statements : (I) $f$ is…
View solution →For which of the following curves, the line $x + \sqrt 3 y = 2\sqrt 3$ is the tangent at the point $\left( {{{3\sqrt 3 } \over 2},{1 \over 2}} \right)$?
View solution →A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is…
View solution →The equation of the normal to the curve y = (1+x)2y + cos 2(sin–1x) at x = 0 is :
View solution →Let $f(x)=\int_0^{x^2} \frac{\mathrm{t}^2-8 \mathrm{t}+15}{\mathrm{e}^{\mathrm{t}}} \mathrm{dt}, x \in \mathbf{R}$. Then the numbers of local maximum and local minimum points of…
View solution →If the curves x = y4 and xy = k cut at right angles, then (4k)6 is equal to __________.
View solution →If the maximum value of $a$, for which the function $f_{a}(x)=\tan ^{-1} 2 x-3 a x+7$ is non-decreasing in $\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)$, is $\bar{a}$, then…
View solution →If Rolle's theorem holds for the function $f(x) = {x^3} - a{x^2} + bx - 4$, $x \in [1,2]$ with $f'\left( {{4 \over 3}} \right) = 0$, then ordered pair (a, b) is equal to :
View solution →Let $A = [{a_{ij}}]$ be a 3 $\times$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {if\,\left| {i - j} \right| = 1} \cr {2x + 1} & , &…
View solution →Let $f:(0,1)\to\mathbb{R}$ be a function defined $f(x) = {1 \over {1 - {e^{ - x}}}}$, and $g(x) = \left( {f( - x) - f(x)} \right)$. Consider two statements (I) g is an increasing…
View solution →Let the normal at a point P on the curve y2 – 3x2 + y + 10 = 0 intersect the y-axis at $\left( {0,{3 \over 2}} \right)$ . If m is the slope of the tangent at P to the curve,…
View solution →Let $f: \rightarrow \mathbb{R} \rightarrow(0, \infty)$ be strictly increasing function such that $\lim _\limits{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1$. Then, the value of…
View solution →$$\text { If } f(x)=\left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{array}\right| \text { for all } x \in \mathbb{R} \text {,…
View solution →Let $(2,3)$ be the largest open interval in which the function $f(x)=2 \log _{\mathrm{e}}(x-2)-x^2+a x+1$ is strictly increasing and (b, c) be the largest open interval, in which…
View solution →If the function $f(x)=2 x^3-9 a x^2+12 \mathrm{a}^2 x+1$, where $\mathrm{a}0$, attains its local maximum and local minimum values at p and q , respectively, such that…
View solution →The shortest distance between the curves $y^2=8 x$ and $x^2+y^2+12 y+35=0$ is:
View solution →Let the function $ f(x) = \frac{x}{3} + \frac{3}{x} + 3, x \neq 0 $ be strictly increasing in $(-\infty, \alpha_1) \cup (\alpha_2, \infty)$ and strictly decreasing in $(\alpha_3,…
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