350. Find the greatest and least values of each of the following functions:
1) $u=x^{3}-3 x^{2}-9 x+35$ on the interval $[-4 ; 4]$;
2) $p=x^{2} \ln x$ on the interval $[1, e]$
3) $r=2 \sin x+\sin 2 x$ on the interval $\left[0 ; \frac{3}{2} \pi\right]$;
4) $y=\operatorname{arctg} x^{2}$.
Solution. According to the practical rule:

1) I. Find the critical points of the function lying within the interval $[-4 ; 4]$, and calculate its values at these points: $u' = 3x^2 - 6x - 9 ; u' = 0$ at points $x = -1$ and $x = 3$. These points lie within the interval $[-4 ; 4]$ and are critical. There are no other critical points, as the derivative $u'$ exists everywhere. The values of the function $u$ at the critical points: $u(-1) = 40 ; u(3) = 8$.

I1. Calculate the values of the function at the endpoints of the interval $[-4 ; 4]$ : $u(-4) = -41 ; u(4) = 15$.

III. Comparing all the calculated values of the function at the internal critical points and at the endpoints of the interval, we conclude: the maximum value of the function $u$ on the interval $[-4 ; 4]$ is 40 and is achieved at the internal critical point $x = -1$, while its minimum value is -41 and is achieved at the left endpoint of the interval $x = -4$ (Fig. 53).

2) I. Find the critical points: $p' = x(1 + 2 \ln x) ; p' = 0$ at points $x_1 = 0$ and $x_2 = e^{-\frac{1}{2}}$. The point $x_1$ lies outside the domain of the given function $0 < x < +\infty$; the point $x_2$ lies outside the given interval $\left[1 ; e\right]$. The derivative $p'$ exists throughout the interval of definition of the function $p$. Therefore, there are no critical points within the given interval.

II. Calculate the values of the function $p$ at the endpoints of the interval: $p(1) = 0 ; p(e) = e^2$.