Question

If \(f\left( 2 \right) = 10\), and \(f'\left( x \right) = {x^2}f\left( x \right)\) for all \(x\), \(f''\left( 2 \right)\)

Short answer

The answer is \(f''\left( 2 \right) = 200\).

Step-by-step solution

Finding derivative when two functions are multiplied

Let the function be \(h\left( x \right) = f\left( x \right)g\left( x \right)\) then the derivative will be,

\(\begin{aligned}h'\left( x \right) & = \frac{d}{{dx}}f\left( x \right)g\left( x \right)\\ & = f\left( x \right)\frac{d}{{dx}}g\left( x \right) + g\left( x \right)\frac{d}{{dx}}f\left( x \right)\end{aligned}\)

Finding the derivative of the function at a given point

Given that \(f'\left( x \right) = {x^2}f\left( x \right)\).

Substitute 2 for x into the function of \(f'\left( x \right)\) to determine \(f'\left( 2 \right)\).

\(\begin{aligned}f'\left( 2 \right) & = 4 \times f\left( 2 \right)\\ & = 40\end{aligned}\).

Differentiating the function we get,

\(\begin{aligned}f''\left( x \right) & = \frac{d}{{dx}}{x^2}f\left( x \right)\\ & = {x^2}f'\left( x \right) + 2xf\left( x \right)\end{aligned}\)

Hence \(f''\left( x \right) = {x^2}f'\left( x \right) + 2xf\left( x \right)\).

Substitute 2 for x into the function of \(f''\left( x \right)\) to determine \(f''\left( 2 \right)\).

\(\begin{aligned}f''\left( 2 \right) & = 4 \times 40 + 2 \times 2 \times 10\\ & = 200\end{aligned}\)

Therefore, \(f''\left( 2 \right) = 200\).