Differentiation Formulas — Question 4

Question 4 - Solution

(a) Differentiate

The quotient and product rules give

$$\boxed{f'(x)=\frac{1-2x\arctan x}{(1+x^2)^2}+e^x x^4(x+5).}$$

(b) Critical numbers

The derivative exists for every real $x$, since $1+x^2>0$.

Thus critical numbers solve $f'(x)=0$. Bisection on $(-5,-4)$ gives

$$\boxed{x\approx -4.995520.}$$

The derivative has opposite signs at the ends of this interval, so continuity guarantees a root there.