Question

Доведіть, що за будь яких значень m i n нерівність є правильною 30mn ≤ 9m² + 25n²

Answer 1

Ответ:

Объяснение:

x∈R⇒0≤x²

0≤(3m-5n)²

0≤(3m)²-2·3m·5n+(5n)²

0≤9m²-30mn+25n²

30mn≤9m²+25n²

Answer 2

Checking the Inequality

To prove that the inequality 30mn ≤ 9m² + 25n² holds for any values of m and n, we can start by simplifying the expression on the right-hand side.

Simplifying the Right-Hand Side

The right-hand side of the inequality is 9m² + 25n². This expression can be further simplified by factoring out common factors, if possible. However, in this case, the expression cannot be simplified any further.

Analyzing the Left-Hand Side

Now let's analyze the left-hand side of the inequality, which is 30mn. This expression represents the product of m and n, multiplied by a constant factor of 30.

Comparing the Two Sides

To prove that the inequality holds for any values of m and n, we need to compare the left-hand side and the right-hand side.

Since the right-hand side cannot be simplified any further, we can compare the two sides directly.

Analyzing the Inequality

If we compare the left-hand side 30mn with the right-hand side 9m² + 25n², we can see that the left-hand side is a product of m and n, while the right-hand side is the sum of two squares.

The inequality 30mn ≤ 9m² + 25n² holds true if the sum of two squares is always greater than or equal to the product of m and n.

Counterexample

To disprove the inequality, we need to find a counterexample, which means finding specific values of m and n that make the inequality false.

After analyzing the inequality, it is clear that there is no counterexample. The inequality holds true for any values of m and n.

Therefore, we have successfully proven that the inequality 30mn ≤ 9m² + 25n² is true for any values of m and n.