Questions: Use the First Derivative Test to find any local extrema. State the location (x-value) of the local maximum. f(x)=-5x^3+15x^2+120x

Solution Steps

To find the local extrema using the First Derivative Test, follow these steps:

1. Compute the first derivative of the function \( f(x) \).
2. Find the critical points by setting the first derivative equal to zero and solving for \( x \).
3. Determine the sign of the first derivative before and after each critical point to identify whether each point is a local maximum, minimum, or neither.

The function is given by

\[ f(x) = -5x^3 + 15x^2 + 120x. \]

The first derivative is calculated as follows:

\[ f'(x) = -15x^2 + 30x + 120. \]

To find the critical points, we set the first derivative equal to zero:

\[ -15x^2 + 30x + 120 = 0. \]

Factoring or using the quadratic formula, we find the critical points:

\[ x = -2 \quad \text{and} \quad x = 4. \]

We analyze the sign of the first derivative around the critical points:

• For \( x = -2 \):

  • \( f'(-3) > 0 \) (left of -2)
  • \( f'(-1) < 0 \) (right of -2)
  • Thus, \( x = -2 \) is a local minimum.

• For \( x = 4 \):

  • \( f'(3) < 0 \) (left of 4)
  • \( f'(5) > 0 \) (right of 4)
  • Thus, \( x = 4 \) is a local maximum.

Final Answer