Questions: f(x)=x^3+7 x^2-16 x-112 Determine the behavior of the function at each zero.

Solution Steps

To determine the behavior of the function at each zero, we first need to find the roots of the polynomial \( f(x) = x^3 + 7x^2 - 16x - 112 \). Once the roots are found, we can analyze the behavior of the function around each root by examining the sign changes in the intervals around the roots. This involves checking the derivative to understand the nature of each zero (whether it is a local minimum, maximum, or a point of inflection).

The roots of the polynomial \( f(x) = x^3 + 7x^2 - 16x - 112 \) are found to be: \[ x = -7, \quad x = -4, \quad x = 4 \]

The derivative of the function is given by: \[ f'(x) = 3x^2 + 14x - 16 \]

We evaluate the derivative at each root to determine the behavior of the function:

1. For \( x = -7 \): \[ f'(-7) = 3(-7)^2 + 14(-7) - 16 = 147 - 98 - 16 = 33 > 0 \quad \text{(local minimum)} \]

2. For \( x = -4 \): \[ f'(-4) = 3(-4)^2 + 14(-4) - 16 = 48 - 56 - 16 = -24 < 0 \quad \text{(local maximum)} \]

3. For \( x = 4 \): \[ f'(4) = 3(4)^2 + 14(4) - 16 = 48 + 56 - 16 = 88 > 0 \quad \text{(local minimum)} \]

Final Answer