"Let $\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}$ and $\beta = \mathop {\min }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}$. If $8{x^2} + bx + c = 0$ is a quadratic equation whose roots are $\alpha$1/5 and $\beta$1/5, then the value of c $-$ b is equal to :"

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Accepted Answer

"$\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}$

$= \max \{ {2^{6\sin 3x}}{.2^{8\cos 3x}}\}$

$= max\{ {2^{6\sin 3x + 8\cos 3x}}\}$

and $$\beta = \min \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\} = \min \{ {2^{6\sin 3x + 8\cos 3x}}\} $$

Now range of $6sin3x + 8cos3x$

$$ = \left[ { - \sqrt {{6^2} + {8^2}} , + \sqrt {{6^2} + {8^2}} } \right] = [ - 10,10]$$

$\alpha$ = 2¹⁰ & $\beta$ = 2^$-$10

So, $\alpha$^1/5 = 2² = 4

$\Rightarrow$ $\beta$^1/5 = 2^$-$2 = 1/4

quadratic 8x² + bx + c = 0

$- {b \over 8} = {{17} \over 4}$ $\Rightarrow$ b = -34

${c \over 8} = 1$ $\Rightarrow$ c = 8

$\therefore$ c – b = 8 + 34 = 42"

Let $\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}$ and $\beta = \mathop {\min }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}$. If $8{x^2} + bx + c = 0$ is a quadratic equation whose roots are $\alpha$^1/5 and $\beta$^1/5, then the value of c $-$ b is equal to :

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Let $\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}$ and $\beta = \mathop {\min }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}$. If $8{x^2} + bx + c = 0$ is a quadratic equation whose roots are $\alpha$1/5 and $\beta$1/5, then the value of c $-$ b is equal to :

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