Question

Let f(x)=x² +5x+8. a. When is the rate of change of f(x) zero? At x= b. On what interval is the derivative positive? (Use I for infinity and - - for negative infinity)

Answer 1

The rate of change of the function f(x) is zero at x = -2.5. The derivative is positive for all x > -2.5, which corresponds to the interval (-2.5, I).

Step-by-step explanation:

To determine when the rate of change of the function f(x) = x² + 5x + 8 is zero, we need to find the derivative of the function and set it equal to zero. The derivative, f'(x), represents the rate of change of the function. Here is how you do it:

• First, find the derivative: f'(x) = 2x + 5.
• Next, set the derivative equal to zero and solve for x: 0 = 2x + 5 leads to x = -5/2.

The rate of change of f(x) is zero at x = -2.5.

For the interval where the derivative is positive, we consider the expression 2x + 5:

• If x > -2.5, then 2x + 5 is positive.
• Therefore, the derivative is positive for the interval (-2.5, I)