Question

Find the relative maximum and relative minimum of the graph of the function.
f(x)=x^3+3x^2 -3x
Determine where the relative maximum occurs and round each coordinate to the nearest hundredth. Choose the correct answer below.
1)(0.07, 10.36)
2)(-1.05,-1.85)
3)(0.41,-0.66)
4)(-2.41, 10.66)

Answer 1

To find the relative maximum and minimum of the function \( f(x) = x^3 + 3x^2 - 3x \), we first need to compute the first derivative and find the critical points:

1. **Find the first derivative \( f'(x) \)**:

\[
f'(x) = \frac{d}{dx}(x^3 + 3x^2 - 3x) = 3x^2 + 6x - 3
\]

2. **Set the first derivative to zero to find critical points**:

\[
3x^2 + 6x - 3 = 0
\]

Dividing the equation by 3:

\[
x^2 + 2x - 1 = 0
\]

We will use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{2^2 - 4(1)(-1)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 4}}{2} = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2}
\]

So the critical points are:

\[
x_1 = -1 + \sqrt{2} \quad \text{and} \quad x_2 = -1 - \sqrt{2}
\]

Calculating the approximate values:

\[
\sqrt{2} \approx 1.414
\]

Thus,

\[
x_1 \approx -1 + 1.414 \approx 0.414 \quad \text{(relative maximum candidate)}
\]
\[
x_2 \approx -1 - 1.414 \approx -2.414 \quad \text{(relative minimum candidate)}
\]

3. **Evaluate the second derivative \( f''(x) \)** to determine concavity:

\[
f''(x) = \frac{d}{dx}(3x^2 + 6x - 3) = 6x + 6
\]

Now evaluate \( f''(x) \) at the critical points:

- For \( x_1 \approx 0.414 \):

\[
f''(0.414) = 6(0.414) + 6 \approx 2.484 + 6 = 8.484 \quad (> 0) \Rightarrow \text{local minimum}
\]

- For \( x_2 \approx -2.414 \):

\[
f''(-2.414) = 6(-2.414) + 6 \approx -14.484 + 6 = -8.484 \quad (< 0) \Rightarrow \text{local maximum}
\]

4. **Find the corresponding function values at the critical points**:

- For \( x_1 \approx 0.414 \):

\[
f(0.414) = (0.414)^3 + 3(0.414)^2 - 3(0.414) \approx 0.071 + 0.513 - 1.242 \approx -0.658 \, \text{(rounded to -0.66)}
\]

- For \( x_2 \approx -2.414 \):

\[
f(-2.414) = (-2.414)^3 + 3(-2.414)^2 - 3(-2.414) \approx -14.094 + 17.516 + 7.242 \approx 10.664 \, \text{(rounded to 10.66)}
\]

5. **Final Results**:

Thus, the coordinates for the relative maximum are:

\(-2.41, 10.66\).

Thus, the correct answer is:

**4)** \((-2.41, 10.66)\)