Find the relative maximum and relative minimum of the graph of the function.

f(x)=x^3+2x^2-5x

Determine where the relative maximum occurs and round each coordinate to the nearest hundredth. Choose the correct answer below.

(0.29,8.94)
(0.79,-2.21)
(-1.16,-0.86)
(-2.12,10.06)

Determine where the relative minimum occurs and round each coordinate to the nearest hundredth. Choose the correct answer below.

(-2.12,10.06)
(0.79,-2.21)
(-1.16,-0.86)
(0.29,8.94)

1 answer

To find the relative maximum and minimum of a function, we need to find the critical points first. The critical points occur when the derivative of the function equals 0 or does not exist.

First, let's find the derivative of the function f(x) = x^3 + 2x^2 - 5x.

f'(x) = 3x^2 + 4x - 5

Next, let's set the derivative equal to 0 and solve for x:

3x^2 + 4x - 5 = 0

We can factor the quadratic equation:

(3x - 1)(x + 5) = 0

Setting each factor equal to 0, we find the critical points:

3x - 1 = 0 --> x = 1/3
x + 5 = 0 --> x = -5

Now, let's plug these critical points back into the original function to find the corresponding y-coordinates.

f(1/3) = (1/3)^3 + 2(1/3)^2 - 5(1/3) = 1/27 + 2/9 - 5/3 = -2.21
f(-5) = (-5)^3 + 2(-5)^2 - 5(-5) = -125 + 50 + 25 = -50

The relative maximum occurs at (0.79, -2.21) and the relative minimum occurs at (-2.12, -50). Neither of these options are listed as choices, so there may be an error in the given answer choices.