Questions: Learning Outcome O5: Problem 5 (1 point) Directions: Use the function f(x) below to answer the following questions. f(x) = x^4 - 50 x^2 - 7 Find the x-values(s) where f has a relative maximum / relative minimum. 1. Relative Maximum(s): 2. Relative Minimum(s):

Solution Steps

To find the relative maximum and minimum of the function \( f(x) = x^4 - 50x^2 - 7 \), we need to follow these steps:

1. Find the derivative: Compute the first derivative of the function, \( f'(x) \), to find the critical points.
2. Solve for critical points: Set \( f'(x) = 0 \) and solve for \( x \) to find the critical points.
3. Determine the nature of critical points: Use the second derivative test by computing \( f''(x) \). Evaluate \( f''(x) \) at each critical point to determine if it is a relative maximum or minimum.

The function is given by \( f(x) = x^4 - 50x^2 - 7 \). We compute the first derivative: \[ f'(x) = 4x^3 - 100x \]

To find the critical points, we set the first derivative equal to zero: \[ 4x^3 - 100x = 0 \] Factoring out \( 4x \), we have: \[ 4x(x^2 - 25) = 0 \] This gives us the critical points: \[ x = -5, \quad x = 0, \quad x = 5 \]

Next, we compute the second derivative: \[ f''(x) = 12x^2 - 100 \] We evaluate the second derivative at each critical point:

• For \( x = -5 \): \[ f''(-5) = 12(-5)^2 - 100 = 300 - 100 = 200 \quad (\text{positive, relative minimum}) \]

• For \( x = 0 \): \[ f''(0) = 12(0)^2 - 100 = -100 \quad (\text{negative, relative maximum}) \]

• For \( x = 5 \): \[ f''(5) = 12(5)^2 - 100 = 300 - 100 = 200 \quad (\text{positive, relative minimum}) \]

Final Answer