Chapter 3 - Section 3.2 - The Product and Quotient Rules - 3.2 Exercises - Page 189: 48

$$f''(2)=200$$

$f(2)=10$ and $f'(x)=x^2f(x)$ for all $x$ Find $f''(x)$ $$f''(x)=(x^2)'f(x)+x^2f'(x)$$ $$f''(x)=2xf(x)+x^2f'(x)$$ $$f''(x)=2xf(x)+x^2(x^2f(x))$$ $$f''(x)=2xf(x)+x^4f(x)$$ So, $$f''(2)=2\times2f(2)+2^4f(2)$$ $$f''(2)=4\times10+16\times10$$ $$f''(2)=200$$