Question

Complete parts​ (a) through​ (c) below.
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 \((0, 8)\).