Question

Find the relative maximum and relative minimum of the graph of the function, f(x) = x^3 + 2x^2 - 2x. Determine where the relative maximum occurs and round each coordinate to the nearest hundredth. Show your work.

(0.39,-0.42)

(-2.68,-1.65)

(-0.75,5.06)

(-1.72,4.27)

Answers

GPT 3.5
To find the relative extrema of the function, we need to find the critical points by taking the derivative of the function and setting it equal to zero.

f(x) = x^3 + 2x^2 - 2x

f'(x) = 3x^2 + 4x - 2

Setting f'(x) = 0:

3x^2 + 4x - 2 = 0

To solve for x, we can use the quadratic formula:

x = (-4 ± √(4^2 - 4(3)(-2))) / (2(3))
x = (-4 ± √(16 + 24)) / 6
x = (-4 ± √40) / 6
x = (-4 ± 2√10) / 6
x = (-2 ± √10) / 3

So the critical points are (-2 + √10) / 3 and (-2 - √10) / 3.

To determine if each critical point is a relative maximum or minimum, we can use the second derivative test. Taking the second derivative:

f''(x) = 6x + 4

Evaluating f'' at each critical point:

f''((-2 + √10) / 3) = 6(-2 + √10) / 3 + 4 = -2 + 2√10 + 4 = 2√10 + 2
f''((-2 - √10) / 3) = 6(-2 - √10) / 3 + 4 = -2 - 2√10 + 4 = -2√10 + 2

Since f''((-2 + √10) / 3) is positive, the point (-2 + √10) / 3 is a relative minimum.
Since f''((-2 - √10) / 3) is negative, the point (-2 - √10) / 3 is a relative maximum.

Now we can evaluate the function at each critical point to find the coordinates of the relative extrema:

f((-2 + √10) / 3) = ((-2 + √10) / 3)^3 + 2((-2 + √10) / 3)^2 - 2(-2 + √10) / 3
f((-2 + √10) / 3) ≈ 5.064

f((-2 - √10) / 3) = ((-2 - √10) / 3)^3 + 2((-2 - √10) / 3)^2 - 2(-2 - √10) / 3
f((-2 - √10) / 3) ≈ -1.647

Rounding to the nearest hundredth, we have the relative maximum at (-2.68, -1.65) and the relative minimum at (-0.75, 5.06). Therefore, the correct answer is:

(-2.68, -1.65)

(-0.75, 5.06)

Related Questions