Question

the graph and equation of the function f are given.
a. use the graph to find any values at which f has a relative maximum, and use
the equation to calculate the relative maximum for each value.
b. use the graph to find any values at which f has a relative minimum, and use
the equation to calculate the relative minimum for each value.

f(x)=2x³+6x²−18x+4

graph with window −5,5,1 by −70,70,10

a. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the function f has (a) relative maxima(maximum) at −3 and the relative maxima(maximum) are(is) 58.
(use a comma to separate answers as needed.)
b. the function f has no relative maxima.

b. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the function f has (a) relative minima(minimum) at □ and the relative minima(minimum) are(is) □.
(use a comma to separate answers as needed.)
b. the function f has no relative minima.

Explanation:

Part a (already solved, but let's verify for context)

To find relative maximum, we can also use calculus (find critical points by taking derivative, then test intervals or use second derivative test). The function is \( f(x) = 2x^3 + 6x^2 - 18x + 4 \). The derivative \( f'(x) = 6x^2 + 12x - 18 \). Set \( f'(x) = 0 \):
\( 6x^2 + 12x - 18 = 0 \) → Divide by 6: \( x^2 + 2x - 3 = 0 \) → Factor: \( (x + 3)(x - 1) = 0 \). So critical points at \( x = -3 \) and \( x = 1 \).

For \( x = -3 \), substitute into \( f(x) \):
\( f(-3) = 2(-3)^3 + 6(-3)^2 - 18(-3) + 4 = 2(-27) + 6(9) + 54 + 4 = -54 + 54 + 54 + 4 = 58 \). So relative maximum at \( x = -3 \), value 58 (matches the given choice A).

Part b: Finding Relative Minimum

Step 1: Identify Critical Point from Derivative

From the derivative \( f'(x) = 6x^2 + 12x - 18 \), we found critical points at \( x = -3 \) (relative max) and \( x = 1 \) (we test this for relative min).

Step 2: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x) = 2x^3 + 6x^2 - 18x + 4 \):
\( f(1) = 2(1)^3 + 6(1)^2 - 18(1) + 4 = 2 + 6 - 18 + 4 = -6 \).

Step 3: Confirm with Graph (or Second Derivative Test)

Second derivative: \( f''(x) = 12x + 12 \). At \( x = 1 \), \( f''(1) = 12(1) + 12 = 24 > 0 \), so concave up (relative minimum).

Answer:

(Part b):
A. The function \( f \) has a relative minimum at \( 1 \) and the relative minimum is \( -6 \).

(So fill in the boxes: first box \( 1 \), second box \( -6 \))