10. (20 points) Find the minimum value of the function

$$
f(x)=\sqrt{(x-a)^{2}+b^{2}}+\sqrt{(x-b)^{2}+a^{2}} \quad a, b>0
$$
Solution:

![](https://cdn.mathpix.com/cropped/2024_05_06_9c64a952bd96df941010g-10.jpg?height=531&width=557&top_left_y=320&top_left_x=795)

We will solve this using a geometric method. Consider three points \( A(a, b), B(b, -a) \), and \( C(x, 0) \). The distance from point \( C \) to point \( A \) is \( \sqrt{(x-a)^{2}+b^{2}} \), and the distance from point \( C \) to point \( B \) is \( \sqrt{(x-b)^{2}+a^{2}} \). Therefore, our function is

\[
f(x)=\sqrt{(x-a)^{2}+b^{2}}+\sqrt{(x-b)^{2}+a^{2}}=C A+C B
\]

the sum of the distances from the variable point \( C \) to points \( A \) and \( B \). By the triangle inequality, \( C A + C B \geqslant A B \), with equality when point \( C \) coincides with point \( D \). We obtain that the minimum value of the function \( f(x) \) is the distance between points \( A \) and \( B \), which is \( \sqrt{(a-b)^{2}+(b+a)^{2}}=\sqrt{2 a^{2}+2 b^{2}} \).

(20 points) Find the minimum value of the function

\[
f(x)=\sqrt{(x+a)^{2}+b^{2}}+\sqrt{(x+b)^{2}+a^{2}} \quad a, b>0
\]

## Solution:

![](https://cdn.mathpix.com/cropped/2024_05_06_9c64a952bd96df941010g-10.jpg?height=543&width=560&top_left_y=1599&top_left_x=791)

We will solve this using a geometric method. Consider three points \( A(-a, b), B(-b, -a) \), and \( C(x, 0) \). The distance from point \( C \) to point \( A \) is \( \sqrt{(x+a)^{2}+b^{2}} \), and the distance from point \( C \) to point \( B \) is \( \sqrt{(x+b)^{2}+a^{2}} \). Therefore, our function is

\[
f(x)=\sqrt{(x+a)^{2}+b^{2}}+\sqrt{(x+b)^{2}+a^{2}}=C A+C B
\]

the sum of the distances from the variable point \( C \) to points \( A \) and \( B \). By the triangle inequality, \( C A + C B \geqslant A B \), with equality when point \( C \) coincides with point \( D \). We obtain that the minimum value of the function \( f(x) \) is the distance between points \( A \) and \( B \), which is \( \sqrt{(a-b)^{2}+(b+a)^{2}}=\sqrt{2 a^{2}+2 b^{2}} \).

Answers:

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