Approxima any relative minimum or maximum values of this function then find the intervals where the function is increasing and decreasing f(x)=-2x^3+5x^2-4

Let's analyze the function \( f(x) = -2x^3 + 5x^2 - 4 \) to approximate any relative minimum or maximum values and find the intervals where the function is increasing and decreasing.

### Step 1: Find the first derivative
The first derivative \( f'(x) \) will help identify critical points and determine the intervals where the function is increasing or decreasing.

\[
f'(x) = \frac{d}{dx}(-2x^3 + 5x^2 - 4) = -6x^2 + 10x
\]

### Step 2: Set the first derivative equal to zero to find critical points
To find critical points, solve \( f'(x) = 0 \):

\[
-6x^2 + 10x = 0
\]

Factor out the common term:

\[
x(-6x + 10) = 0
\]

This gives us two solutions:

\[
x = 0 \quad \text{or} \quad -6x + 10 = 0
\]

Solving \( -6x + 10 = 0 \):

\[
x = \frac{10}{6} = \frac{5}{3}
\]

So the critical points are \( x = 0 \) and \( x = \frac{5}{3} \).

### Step 3: Determine the sign of \( f'(x) \) to find intervals of increasing and decreasing behavior
We will test the sign of \( f'(x) = -6x^2 + 10x \) in the intervals divided by the critical points: \( (-\infty, 0) \), \( (0, \frac{5}{3}) \), and \( (\frac{5}{3}, \infty) \).

- For \( x \in (-\infty, 0) \), choose \( x = -1 \):
  \[
  f'(-1) = -6(-1)^2 + 10(-1) = -6 - 10 = -16 \quad (\text{negative, so } f(x) \text{ is decreasing})
  \]

- For \( x \in (0, \frac{5}{3}) \), choose \( x = 1 \):
  \[
  f'(1) = -6(1)^2 + 10(1) = -6 + 10 = 4 \quad (\text{positive, so } f(x) \text{ is increasing})
  \]

- For \( x \in (\frac{5}{3}, \infty) \), choose \( x = 2 \):
  \[
  f'(2) = -6(2)^2 + 10(2) = -24 + 20 = -4 \quad (\text{negative, so } f(x) \text{ is decreasing})
  \]

### Step 4: Identify relative minimum and maximum
- At \( x = 0 \), the function changes from decreasing to increasing, so there is a **relative minimum** at \( x = 0 \).
- At \( x = \frac{5}{3} \), the function changes from increasing to decreasing, so there is a **relative maximum** at \( x = \frac{5}{3} \).

### Step 5: Approximate the relative minimum and maximum values
To approximate the values of the relative minimum and maximum, we substitute the critical points into the original function \( f(x) \):

- At \( x = 0 \):
  \[
  f(0) = -2(0)^3 + 5(0)^2 - 4 = -4
  \]
  So, the relative minimum value is \( -4 \) at \( x = 0 \).

- At \( x = \frac{5}{3} \):
  \[
  f\left(\frac{5}{3}\right) = -2\left(\frac{5}{3}\right)^3 + 5\left(\frac{5}{3}\right)^2 - 4 = -\frac{250}{27} + \frac{125}{9} - 4 \approx 1.46
  \]
  So, the relative maximum value is approximately \( 1.46 \) at \( x = \frac{5}{3} \).

### Step 6: Summary of increasing and decreasing intervals
- **Increasing** on \( (0, \frac{5}{3}) \)
- **Decreasing** on \( (-\infty, 0) \) and \( (\frac{5}{3}, \infty) \)

### Final Result
- Relative minimum at \( x = 0 \), \( f(0) = -4 \)
- Relative maximum at \( x = \frac{5}{3} \), \( f\left(\frac{5}{3}\right) \approx 1.46 \)
- Increasing on \( (0, \frac{5}{3}) \)
- Decreasing on \( (-\infty, 0) \) and \( (\frac{5}{3}, \infty) \)

---

Would you like more details on any part of the process?

Here are some related questions you might explore:
1. How do higher-order derivatives affect the concavity of a function?
2. How can we apply the second derivative test to verify maximum and minimum points?
3. What are inflection points, and how can they be found for a cubic function?
4. How do we solve cubic equations algebraically to find exact roots?
5. How does the degree of a polynomial affect the number of critical points?

**Tip:** The first derivative test is a simple way to check for relative extrema, but the second derivative test can often confirm concavity more efficiently.

Analyze the function f(x) = -2x^3 + 5x^2 - 4 to find its relative minimum and maximum points and identify the intervals where the function is increasing and decreasing. Learn how to use the first derivative to solve calculus problems step by step. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation