Example 7. What dimensions of a box (without a lid) made from a square sheet of cardboard with side $a$ will have the maximum capacity
Solution. To manufacture a box, it is necessary to cut out squares from the corners of the sheet and fold the protrusions of the resulting cross-shaped figure. Let the side of the cut-out square be denoted by $x$, then the side of the box base will be $a-2 x$. The volume of the box can be expressed by the function $V=(a-2 x)^{2} x=\left(a^{2}-4 a x+4 x^{2}\right) x=a^{2} x-4 a x^{2}+4 x^{3}$, which is defined in the interval $\left[0, \frac{a}{2}\right]$.

Differentiating the function $V$, we get:

$$
V^{\prime}=a^{2}-8 a x+12 x^{2} ; V^{\prime \prime}=-8 a+24 x
$$

$V^{\prime}=0$, when $a^{2}-8 a x+12 x^{2}=0$, from which

$$
x_{1}=\frac{a}{2}, x_{2}=\frac{a}{6}
$$

Substituting the obtained values of $x$ into the expression for the second derivative:

$$
V^{\prime \prime}\left(\frac{a}{2}\right)=-8 a+12 a=4 a>0 ; \quad V^{\prime \prime}\left(\frac{a}{6}\right)=-8 a+4 a=-4 a<0
$$

Therefore, the maximum volume will be at $x=\frac{a}{6}$. The base of the box is a square with side $\frac{2}{3} a$.

## Problems

Determine the intervals of increase and decrease of the functions:

1. $y=x^{2}-4 x+4$.
2. $y=6-3 x^{2}-x^{3}$.
3. $y=e^{x}$.
4. $y=x^{4}-2 x^{2}$.

Investigate the functions for extrema:
5. $y=x^{3}+6 x^{2}+9 x+2$.
6. $y=\frac{\left(1-x^{2}\right)^{2}}{4}$.
7. $y=\frac{a}{2}\left(e^{\frac{x}{a}}-e^{-\frac{x}{a}}\right)$. 8. $y=\frac{\ln x}{x}$.

Find the maximum and minimum values of the functions in the interval $[-2,2]:$
9. $y=-x^{2}$.
10. $y=-x^{3}$.

Find the maximum and minimum values of the functions:
11. $y=\frac{x^{2}}{1+x^{2}}$.
12. $y=\frac{1}{1+x^{2}}$.

13. Among all rectangles with a given perimeter $2 a$, find the one with the largest area.
14. Which of the right-angled triangles with a given perimeter $2 p$ has the largest area?
15. The lateral sides and the smaller base of the trapezoid are equal to 10 cm. Determine its larger base so that the area of the trapezoid is the largest.
16. In a given ellipse, inscribe a rectangle of the largest area with sides parallel to the axes of the ellipse.
17. An open vessel consists of a cylinder ending at the bottom with a hemisphere; the thickness of the walls is constant. What should be the dimensions of the vessel so that, for a given capacity, the least amount of material is used for its manufacture?

## Answers

1. Decreases in the interval $(-\infty, 2)$, increases in the interval $(2, \infty)$. 2. Decreases in the intervals $(-\infty,-2)$ and $(0, \infty)$, increases in $(-2,0)$. 3. Increases in the interval $(-\infty, \infty)$. 4. Decreases in the intervals $(-\infty,-1)$ and $(0,1)$, increases in the intervals $(-1,0)$ and $(1, \infty) .5 . \min f(x)=f(-1)=$ $=-2, \quad \max f(x)=f(-3)=2 . \quad$ 6. $\max f(x)=f(0)=\frac{1}{4}, \quad \min f(x)=$ $=f(-1)=f(1)=0.7$. No extremum. 8. $\max f(x)=f(e)=\frac{1}{e} \approx 0,37.9 .0$;

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-125.jpg?height=79&width=1085&top_left_y=1461&top_left_x=45)

14. Isosceles. 15. 20 cm. 16. $a \sqrt{2}, b \sqrt{2}$, where $a$ and $b$ are the semi-axes of the ellipse.

## § 3.4. Convexity and Concavity of a Curve. Points of Inflection. Asymptotes of a Curve.

The graph of the function $y=f(x)$ is called concave up (or convex down) in the interval $(a, b)$ if the corresponding arc of the curve lies above the tangent at any point $M[x, f(x)]$ of this arc (Fig. 3.11).

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-126.jpg?height=400&width=487&top_left_y=330&top_left_x=43)

Fig. 3.11

The graph of the function $y=f(x)$ is called concave down (or convex up) in the interval $(a, b)$ if the corresponding arc of the curve lies below the tangent drawn at any point $M[x, f(x)]$ of this arc (Fig. 3.12).

The sufficient condition for concavity (convexity) of a curve: if the second derivative of the function $y=f(x)$ is positive inside the interval $(a, b)$, then the graph of the function is concave up in this interval; if the second derivative $f^{\prime \prime}(x)$ is negative inside the interval $(a, b)$, then the graph of the function $y=f(x)$ is concave down in this interval. (Compare these conditions with the second rule for finding extrema).

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-126.jpg?height=412&width=482&top_left_y=1048&top_left_x=35)

Fig. 3.12

A point of inflection of a continuous curve $y=f(x)$ is a point $M_{0}\left[x_{0}, f\left(x_{0}\right)\right]$, at which the curve changes its concavity to convexity or vice versa (relative to, for example, the downward direction). For the abscissa of the point of inflection $x_{0}$ of the graph of the function $y=f(x)$, the second derivative $f^{\prime \prime}\left(x_{0}\right)$ is zero or does not exist. Points where $f^{\prime \prime}\left(x_{0}\right)=0$ or $f^{\prime \prime}\left(x_{0}\right)$ does not exist are called critical points of the second kind.

If the second derivative of the function $y=f(x)$ changes sign when passing through the critical point $x_{0}$, then $M\left[x_{0}, f\left(x_{0}\right)\right]$ is a point of inflection.

## Asymptotes of a Curve

1. An asymptote of a curve is a straight line to which a point on the curve approaches indefinitely as it moves away from the origin.
2. If

$$
\lim _{x \rightarrow a} f(x)=\infty
$$

then the line $x=a$ is a vertical asymptote of the curve $y=f(x)$.

3. If the limits exist:

$$
\lim _{x \rightarrow+\infty} \frac{f(x)}{x}=k_{1}, \lim _{x \rightarrow+\infty}\left[f(x)-k_{1} x\right]=b_{1}
$$

or

$$
\lim _{x \rightarrow-\infty} \frac{f(x)}{x}=k_{2}, \lim _{x \rightarrow-\infty}\left[f(x)-k_{2} x\right]=b_{2}
$$

then the line $y=k_{1} x+b_{1}$ is the right oblique asymptote of the curve $y=f(x)$, and the line $y=k_{2} x+b_{2}$ is the left oblique asymptote.

4. If in the right-hand side of the equation of the curve $y=f(x)$, a linear part can be separated

$$
y=f(x)=k x+b+\alpha(x)
$$

where $\alpha(x) \rightarrow 0$ as $x \rightarrow \pm \infty$, then the line $y=k x+b$ is an asymptote.