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)=2x^3 + 6x^2 - 18x + 4 ) (-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. (\bigcirc) a. the function ( f ) has (a) relative maxima(maximum) at (square) and the relative maxima(maximum) are(is) (square). (use a comma to separate answers as needed.) (\bigcirc) b. the function ( f ) has no relative maxima.

Explanation:

Step1: Find the derivative of \( f(x) \)

To find relative extrema, we first find the derivative of \( f(x) = 2x^3 + 6x^2 - 18x + 4 \). Using the power rule, the derivative \( f'(x) \) is:
\( f'(x) = 6x^2 + 12x - 18 \)

Step2: Set the derivative equal to zero and solve for \( x \)

To find critical points, set \( f'(x) = 0 \):
\( 6x^2 + 12x - 18 = 0 \)
Divide both sides by 6:
\( x^2 + 2x - 3 = 0 \)
Factor the quadratic equation:
\( (x + 3)(x - 1) = 0 \)
So, the critical points are \( x = -3 \) and \( x = 1 \).

Step3: Analyze the graph or use the second derivative test (or first derivative test) to determine relative maxima/minima

From the graph (or by analyzing the sign changes of \( f'(x) \)):

• For \( x = -3 \): The function changes from increasing to decreasing (so it's a relative maximum).
• For \( x = 1 \): The function changes from decreasing to increasing (so it's a relative minimum).

Step4: Calculate the relative maximum at \( x = -3 \)

Substitute \( x = -3 \) into \( f(x) \):
\( f(-3) = 2(-3)^3 + 6(-3)^2 - 18(-3) + 4 \)
\( = 2(-27) + 6(9) + 54 + 4 \)
\( = -54 + 54 + 54 + 4 \)
\( = 58 \)

Answer:

a. The function \( f \) has a relative maximum at \( \boldsymbol{-3} \) and the relative maximum is \( \boldsymbol{58} \). So the correct choice is A, with the first box as \(-3\) and the second box as \(58\).