Complete parts​ (a) through​ (c) below.

Question

a. Graph y equals x cubed plus 5 x squared plus 8 on a window that shows a local minimum and a local maximum.
b. A local maximum occurs at what​ point?
c. A local minimum occurs at what​ point?
Question content area bottom
Part 1
a. Choose the correct graph below.
A.

A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from 0 to 30 in increments of 1. A smooth curve rises at a decreasing rate to a maximum in the second quadrant, falls to a minimum, then rises at an increasing rate in the first quadrant.
​[minus​5,5,1] by​ [0,30,1]
Your answer is correct.B.

A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from 0 to 30 in increments of 1. A smooth curve falls at a decreasing rate in the second quadrant to a minimum, rises to a maximum in the first quadrant, then falls at an increasing rate.
​[minus​5,5,1] by​ [0,30,1]
C.

A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from 0 to 30 in increments of 1. A smooth curve falls at a decreasing rate to a minimum in the third quadrant, rises to a maximum, then falls at an increasing rate in the fourth quadrant.
​[minus​5,5,1] by ​[minus​30,0,1]
D.

A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from 0 to 30 in increments of 1. A smooth curve rises at a decreasing rate in the third quadrant to a maximum, falls to a minimum in the fourth quadrant, then rises at an increasing rate.
​[minus​5,5,1] by ​[minus​30,0,1]
Part 2
b. A local maximum occurs at
  
enter your response here.
​(Type an ordered​ pair, using integers or decimals. Round to the nearest hundredth as​ needed.)

Answer 1

To analyze the function \(y = x^3 + 5x^2 + 8\) for local maximums and minimums, we first need to find its critical points by calculating the first derivative and setting it to zero.

Step 1: Find the Derivative

The function's derivative is found as follows:

\[ y' = 3x^2 + 10x \]

Step 2: Set the Derivative to Zero

To find critical points, set the first derivative equal to zero:

\[ 3x^2 + 10x = 0 \]

Step 3: Factor the Equation

Factoring gives:

\[ x(3x + 10) = 0 \]

This gives us:

\[ x = 0 \quad \text{or} \quad 3x + 10 = 0 \implies x = -\frac{10}{3} \approx -3.33 \]

Step 4: Determine Local Maximum and Minimum

To find out whether these critical points are local maxima or minima, we can use the second derivative test.

The second derivative is:

\[ y'' = 6x + 10 \]

Evaluate the second derivative at each critical point:

1. At \(x = 0\):

\[ y''(0) = 6(0) + 10 = 10 > 0 \implies \text{local minimum} \]

2. At \(x = -\frac{10}{3}\):

\[ y''\left(-\frac{10}{3}\right) = 6\left(-\frac{10}{3}\right) + 10 = -20 + 10 = -10 < 0 \implies \text{local maximum} \]

Step 5: Find Corresponding y-Values

Now calculate the y-values for these x-values:

At \(x = 0\): \[ y(0) = 0^3 + 5(0)^2 + 8 = 8 \] Thus, the local minimum is at \((0, 8)\).

At \(x = -\frac{10}{3}\): \[ y\left(-\frac{10}{3}\right) = \left(-\frac{10}{3}\right)^3 + 5\left(-\frac{10}{3}\right)^2 + 8 \] Calculating this: \[ = -\frac{1000}{27} + 5 \cdot \frac{100}{9} + 8 = -\frac{1000}{27} + \frac{500}{9} + 8 = -\frac{1000}{27} + \frac{1500}{27} + \frac{216}{27} = \frac{-1000 + 1500 + 216}{27} = \frac{716}{27} \approx 26.52 \] Thus, the local maximum occurs at approximately \(\left(-\frac{10}{3}, \frac{716}{27}\right)\) or \((-3.33, 26.52)\).

Step 6: Summarize Results

Part 1 (Graph): You would select the graph that shows a local maximum and minimum, typically graph A based on the upward and downward curves indicated.

Part 2:

• A local maximum occurs at approximately \((-3.33, 26.52)\).
• A local minimum occurs at \((0, 8)\).

Therefore, your answers are:

b. The local maximum occurs at \((-3.33, 26.52)\).

c. The local minimum occurs at \((0, 8)\).