Example 2.26. Find the intervals of monotonicity of the function $y=x^{4}-\frac{4}{3} x^{3}+5$.
Solution. The domain of the given function is the entire number line. Differentiating, we find

$$
y^{\prime}=4 x^{3}-4 x^{2}=4 x^{2}(x-1) .
$$

The derivative $y^{\prime}$ has no points of discontinuity. The zeros of the derivative $y^{\prime}$ are the roots of the equation $x^{2}(x-1)=0$, i.e., $x_{1}=0$ and $x_{2}=1$.

The domain of the function - the $\mathrm{Ox}$ axis - is divided by these points into three intervals $(-\infty ; 0),(0 ; 1),(1 ;+\infty)$, in each of which $y^{\prime}$ maintains a certain sign. Substituting into the expression for $y^{\prime}$ the values $x=-5, x=\frac{1}{2}, x=5$ from these intervals, we get the signs «-», «-», «+» respectively. Therefore, the function decreases in the interval $(-\infty ; 1)$ and increases in the interval $(1 ;+\infty)$.

A point $x=x_{0}$ is called a point of maximum (minimum) of the function $y=f(x)$ if there exists a neighborhood of the point $x_{0}$ such that for all $x\left(x \neq x_{0}\right)$ in this neighborhood, the inequality

$$
f(x)f\left(x_{0}\right)\right] .
$$

holds.

Points of maximum and minimum of a function are called its points of extremum, and the value of the function at a point of maximum (minimum) is called the maximum (minimum) or extremum of the function.

Points of extremum can only be critical points of the first kind, i.e., points belonging to the domain of the function where the first derivative $f^{\prime}(x)$ is zero or has a discontinuity.

Points of extremum are only those critical points where the first derivative $f^{\prime}(x)$ changes sign when passing through them, and specifically, if the sign of $f^{\prime}(x)$ changes from «+» to «-» (from «-» to «+») when passing through the critical point $x=x_{0}$ in the positive direction, then the point $x=x_{0}$ is a point of maximum (minimum).

From this, we obtain the rule for finding the extrema of the function $y=f(x)$:

1. Find the zeros and points of discontinuity of $f^{\prime}(x)$.
2. Determine the sign of $f^{\prime}(x)$ in the intervals into which the points obtained in step 1 divide the domain of the function $f(x)$ by the method of trial.
3. Select from these points those at which the function $f(x)$ is defined and on either side of which the derivative $f(x)$ has different signs - these are the extremal points. In this case, the extremal point $x=x_{0}$ is a point of maximum if, when moving along the $O x$ axis in the positive direction, it separates an interval where the derivative $f^{\prime}(x)>0$ from an interval where $f\left(x_{0}\right)0$.