In this task, you're asked to find:

1. The minimum value of the expression $a + b$, given that $a \cdot b = 100$;
2. The maximum value of the expression $a \cdot b$, given that $a + b = 100$.

The Cauchy inequality (Коші) provided can help, which is written as:

$\frac{a + b}{2} \geq \sqrt{ab}$

This simplifies to:

$a + b \geq 2\sqrt{ab}$

Let’s use these steps to approach each part:

1) Finding the minimum value of $a + b$ (given $a \cdot b = 100$)

By applying the inequality:

$a + b \geq 2 \sqrt{ab}$

Substitute $ab = 100$:

$a + b \geq 2 \sqrt{100} = 20$

Therefore, the minimum value of $a + b$ is 20, and equality holds when $a = b$.

So, $a = b$ gives:

$a^2 = 100 \quad \Rightarrow \quad a = b = 10$

Thus, the minimum value of $a + b$ is 20 when $a = 10$ and $b = 10$.

2) Finding the maximum value of $a \cdot b$ (given $a + b = 100$)

We need to maximize the product $ab$ under the condition $a + b = 100$. This is a standard optimization problem, and the product $ab$ is maximized when $a = b$.

So, if $a + b = 100$, let $a = b$. Then:

$2a = 100 \quad \Rightarrow \quad a = 50$

Thus, $ab = 50 \times 50 = 2500$.

Therefore, the maximum value of $ab$ is 2500 when $a = b = 50$.

Summary:

1. The minimum value of $a + b$ is 20 when $a = 10$ and $b = 10$.
2. The maximum value of $ab$ is 2500 when $a = 50$ and $b = 50$.

Would you like a more detailed explanation of any part? Here are five related questions you could explore:

1. How do you derive the Cauchy inequality in this context?
2. Why is $a = b$ optimal for both minimizing and maximizing in these cases?
3. How does the geometric mean relate to this problem?
4. Can you solve similar problems using the Arithmetic-Geometric Mean Inequality (AGM)?
5. How would the solutions change if the product $ab$ were negative?

Tip: The Cauchy-Schwarz inequality is a powerful tool for solving optimization problems involving sums and products, especially when equality cases provide critical insights!