Example 3. Point $M$ lies inside a triangle, $k_{1}, k_{2}, k_{3}$ are the distances from $M$ to the sides of the triangle, $h_{1}, h_{2}, h_{3}$ are the corresponding heights. Find the minimum value of the expression

$$
\left(\frac{k_{1}}{h_{1}}\right)^{\alpha}+\left(\frac{k_{2}}{h_{2}}\right)^{\alpha}+\left(\frac{k_{3}}{h_{3}}\right)^{\alpha} \quad \text { for } \alpha \geqslant 1
$$
S o l u t i o n. We have:

$$
2 S=a k_{1}+b k_{2}+c k_{3}=a h_{1}=b h_{2}=c h_{3}
$$

where $S$ is the area of the triangle.

Divide both sides of the equality $a k_{1}+b k_{2}+c k_{3}=a h_{1}$ by $a h_{1}$:

$$
\frac{k_{1}}{h_{1}}+\frac{b}{a} \cdot \frac{k_{2}}{h_{1}}+\frac{c}{a} \cdot \frac{k_{3}}{h_{1}}=1
$$

Since $\frac{b}{a}=\frac{h_{1}}{h_{2}}$ and $\frac{c}{a}=\frac{h_{1}}{h_{3}}$,

then

$$
\frac{k_{1}}{h_{1}}+\frac{k_{2}}{h_{2}}+\frac{k_{3}}{h_{3}}=1
$$

By the monotonicity property of the power mean, we get:

$$
\left(\frac{k_{1}}{h_{1}}\right)^{\alpha}+\left(\frac{k_{2}}{h_{2}}\right)^{\alpha}+\left(\frac{k_{3}}{h_{3}}\right)^{\alpha} \geqslant 3\left(\frac{1}{3}\right)^{\alpha}=\frac{1}{3^{\alpha-1}} \text { for } \alpha \geqslant 1 .
$$

Thus, the minimum value of this expression is $\frac{1}{3^{\alpha-1}}$.

## EXERCISES

119. Find the maximum values of the functions:

a) $y=x^{n}(a-x), 00$.

120. Find the minimum values of the functions:
a) $y=x^{2}(1+x),-10$.
125. Find the maximum values of the functions:

$$
u=\sum_{i=1}^{n}\left(x_{i}+y_{i}\right)^{2}, v=\sum_{i=1}^{n}\left(x_{i}-y_{i}\right)^{2}
$$

if

$$
\sum_{i=1}^{n} x_{i}^{2}=a^{2}, \quad \sum_{i=1}^{n} y_{i}^{2}=b^{2}
$$

126. Find the maximum and minimum values of the function

$$
u=\sum_{i=1}^{k}\left(1+x_{i}\right)^{n}+\sum_{i=1}^{k}\left(1-x_{i}\right)^{n}
$$

where $n, k \in \mathbf{N},-1 \leqslant x_{i} \leqslant 1$.

127. Find the minimum value of the function

$$
y=\left(\frac{1+\sin ^{2} x}{\sin ^{2} x}\right)^{m}+\left(\frac{1+\cos ^{2} x}{\cos ^{2} x}\right)^{m}
$$

where $m \in \mathbf{N}$.

128. Find the minimum value of the function $y=\sqrt{x^{2}-2 m x+n^{2}}+\sqrt{x^{2}-2 p x+q^{2}}, \quad 0<m<n, \quad 0<p<q$.
129. For what value of $x$ does the product $(1-x)^{4}\left(1-x^{2}\right) \times$ $\times(1+2 x)^{2}$ reach its maximum value and what is this value?
130. Find the maximum value of the function

$$
u=\frac{\left(a_{1} x_{1}+a_{2} x_{2}+\ldots+a_{n} x_{n}\right)^{2}}{x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}}
$$

131. Find the minimum value of the function

$$
u=\frac{\left(x_{1}+x_{2}+\ldots+x_{n}\right)^{\frac{n(n+1)}{2}}}{x_{1} x_{2}^{2} x_{3}^{3} \ldots x_{n}^{n}}
$$

132. Two lanterns hang at points $A$ and $B$, one with a light intensity of $J$ candles, and the other with 8 times greater. Find the least illuminated point on $[A B]$, if $|A B|=l$.
133. Through an internal point $P$ of triangle $A B C$, three segments are drawn: $\left[P A_{1}\right]\left\|[B C]\left(A_{1} \in[A C]\right),\left[P B_{1}\right]\right\|[A C]\left(B_{1} \in[A B]\right)$, $\left[P C_{1}\right] \|[A B]\left(C_{1} \in[B C]\right)$. Find the minimum values of the expressions for $\alpha \geqslant 1$ :

$$
\left(\frac{\left|P A_{1}\right|}{a}\right)^{\alpha}+\left(\frac{\left|P B_{1}\right|}{b}\right)^{\alpha}+\left(\frac{\left|P C_{1}\right|}{c}\right)^{\alpha}, \quad\left(\frac{a}{\left|P A_{1}\right|}\right)^{\alpha}+\left(\frac{b}{\left|P B_{1}\right|}\right)^{\alpha}+\left(\frac{c}{\left|P C_{1}\right|}\right)^{\alpha}
$$

134. Let $x, y, z$ be the distances from the center of the inscribed circle of triangle $A B C$ to the vertices. Find the minimum value of the ratio

$$
\frac{a^{\alpha} x^{2 \alpha}+b^{\alpha} y^{2 \alpha}+c^{\alpha} z^{2 \alpha}}{a^{\alpha} b^{\alpha} c^{\alpha}} \text { for } \alpha \geqslant 1
$$

135. In a given segment, inscribe a rectangle of maximum area such that one side lies on the chord.
136. Through an internal point of a triangle of area $S$, lines are drawn parallel to the sides. $S_{1}, S_{2}, S_{3}$ are the areas of the resulting triangles. Find the minimum value of the ratio $\left(S_{1}^{\alpha}+S_{2}^{\alpha}+S_{3}^{\alpha}\right): S^{\alpha}$ for $\alpha \geqslant \frac{1}{2}$.
137. Find the minimum value of the expression

$$
\left(\frac{x}{a}\right)^{2 n}+\left(\frac{y}{b}\right)^{2 n}+\left(\frac{z}{c}\right)^{2 n}
$$

where $a, b, c$ are the sides of a triangle, and $x, y, z$ are the corresponding sides of a triangle with vertices at the feet of the altitudes of the given triangle.

Find the maximum value of the expression

$$
\left(\frac{x}{a}\right)^{\alpha}+\left(\frac{y}{b}\right)^{\alpha}+\left(\frac{z}{c}\right)^{\alpha} \quad \text { for } \alpha \leqslant 1
$$

138. Through an