Question

Докажите что выражение 2x⁶ + (4 - x²)² - 16 не может принимать отрицательных значений​

Answer 1

Объяснение:

упростим:

2$x^{6}$ + 16 - 8$x^{2}$ + $x^{4}$ - 16

2$x^{6}$ + $x^{4}$ - 8$x^{2}$

Доказать невозможно, например при x = 0.01 - выражение отрицательно

Answer 2

Proof that the Expression 2x⁶ (4 - x²)² - 16 Cannot Take Negative Values

To prove that the expression 2x⁶ (4 - x²)² - 16 cannot take negative values, we can analyze the factors and properties of the expression.

Step 1: Factorization First, let's factorize the given expression to understand its components and properties.

The given expression is: 2x⁶ (4 - x²)² - 16

Step 2: Analyzing the Factors We can break down the expression into its constituent factors and analyze their properties to understand the behavior of the entire expression.

- The factor 2x⁶ is always non-negative for real values of x. - The factor (4 - x²)² is also always non-negative for real values of x, as it is squared. - The constant term -16 is always negative.

Step 3: Combining the Factors When we combine the factors, we get the expression 2x⁶ (4 - x²)² - 16. Since the first two factors are always non-negative, the only part that can potentially make the entire expression negative is the constant term -16.

Step 4: Analyzing the Constant Term For the entire expression to be negative, the constant term -16 would have to outweigh the non-negative values of the other factors. However, given that the other factors are always non-negative, the constant term -16 alone cannot make the entire expression negative.

Therefore, the expression 2x⁶ (4 - x²)² - 16 cannot take negative values.

This conclusion is based on the analysis of the factors and properties of the given expression.