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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!

Find the minimum value of a + b, given that a * b = 100, and the maximum value of a * b, given that a + b = 100.

Learn how to solve a problem using the Cauchy inequality to find the minimum of a + b when a * b = 100, and the maximum of a * b when a + b = 100. This exercise involves algebraic optimization and inequality concepts, suitable for Grades 9-12.

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