"Suppose a differentiable function f(x) satisfies the identity f(x+y) = f(x) + f(y) + xy2 + x2y, for all real x and y. $\mathop {\lim }\limits_{x \to 0} {{f\left( x \right)} \over x} = 1$, then f'(3) is equal to ______."

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"Given, f(x + y) = f(x) + f(y) + xy² + x²y ...(1)

differentiating partially with respect to x,

f'(x+y) = f'(x) + 0 + y² + y(2x) [y = constant]

Put x = 0 and y = x

$\therefore$ f'(x) = f'(0) + x² ....(2)

putting x = y = 0 at equation (1),

f(0) = 2f(0)

$\Rightarrow$ f(0) = 0

Given, $\mathop {\lim }\limits_{x \to 0} {{f(x)} \over x} = 1$

This is in $\frac{0}{0}$ form, so we can apply L' hospital rule.

$\mathop {\lim }\limits_{x \to 0} {{f'(x)} \over 1} = 1$

$\Rightarrow f'(0) = 1$

Putting value of f'(0) at equation (2), we get

f'(x) = 1 + x²

$\therefore$ f'(3) = 1 + 3² = 10"

Suppose a differentiable function f(x) satisfies the identity
f(x+y) = f(x) + f(y) + xy² + x²y, for all real x and y.
$\mathop {\lim }\limits_{x \to 0} {{f\left( x \right)} \over x} = 1$, then f'(3) is equal to ______.

Solution

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Suppose a differentiable function f(x) satisfies the identity f(x+y) = f(x) + f(y) + xy2 + x2y, for all real x and y. $\mathop {\lim }\limits_{x \to 0} {{f\left( x \right)} \over x} = 1$, then f'(3) is equal to ______.

Solution

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More Limits, Continuity and Differentiability questions

Drill 25 more like these. Every day. Free.

Suppose a differentiable function f(x) satisfies the identity
f(x+y) = f(x) + f(y) + xy² + x²y, for all real x and y.
$\mathop {\lim }\limits_{x \to 0} {{f\left( x \right)} \over x} = 1$, then f'(3) is equal to ______.