Math Problem Statement

Consider the functionf(x)=x33x2+2f(x)=x^3-3x^2+2. Determine where the function is increasing or decreasing.

Solution

To determine where the function f(x)=x33x2+2f(x) = x^3 - 3x^2 + 2 is increasing or decreasing, we need to analyze its first derivative f(x)f'(x). Here's the step-by-step process:

1. Find the first derivative

The first derivative of f(x)f(x) represents the rate of change of the function and will help us determine intervals of increase or decrease.

f(x)=ddx(x33x2+2)f'(x) = \frac{d}{dx} \left( x^3 - 3x^2 + 2 \right)

Using basic differentiation rules:

f(x)=3x26xf'(x) = 3x^2 - 6x

2. Set the first derivative equal to zero

To find the critical points (where the derivative is zero), set f(x)=0f'(x) = 0:

3x26x=03x^2 - 6x = 0

Factor the equation:

3x(x2)=03x(x - 2) = 0

So, x=0x = 0 or x=2x = 2 are the critical points.

3. Determine the sign of f(x)f'(x) on intervals

Now, we need to test the sign of f(x)f'(x) on the intervals determined by the critical points x=0x = 0 and x=2x = 2. These intervals are:

  • (,0)(-\infty, 0)
  • (0,2)(0, 2)
  • (2,)(2, \infty)

To do this, choose a test point in each interval and substitute it into f(x)=3x26xf'(x) = 3x^2 - 6x:

  • For x(,0)x \in (-\infty, 0), pick x=1x = -1: f(1)=3(1)26(1)=3+6=9>0f'(-1) = 3(-1)^2 - 6(-1) = 3 + 6 = 9 > 0 So, the function is increasing on (,0)(-\infty, 0).

  • For x(0,2)x \in (0, 2), pick x=1x = 1: f(1)=3(1)26(1)=36=3<0f'(1) = 3(1)^2 - 6(1) = 3 - 6 = -3 < 0 So, the function is decreasing on (0,2)(0, 2).

  • For x(2,)x \in (2, \infty), pick x=3x = 3: f(3)=3(3)26(3)=2718=9>0f'(3) = 3(3)^2 - 6(3) = 27 - 18 = 9 > 0 So, the function is increasing on (2,)(2, \infty).

4. Summary of intervals

  • The function is increasing on (,0)(-\infty, 0) and (2,)(2, \infty).
  • The function is decreasing on (0,2)(0, 2).

5. Critical Points and Behavior

At x=0x = 0, the function changes from increasing to decreasing, so x=0x = 0 is a local maximum.

At x=2x = 2, the function changes from decreasing to increasing, so x=2x = 2 is a local minimum.


Would you like any further details or clarification? Here are some related questions you might find interesting:

  1. How to find the second derivative of f(x)f(x) and interpret concavity?
  2. How to determine the local maximum and minimum values of the function?
  3. How can you find the inflection points of the function?
  4. How to analyze the global behavior of the function f(x)f(x)?
  5. What is the behavior of f(x)f(x) as xx \to \infty or xx \to -\infty?

Tip: The first derivative test is a powerful tool to determine where a function is increasing or decreasing. Checking the sign of the derivative in different intervals gives insight into the function's behavior without needing to graph it.

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Critical Points
Increasing and Decreasing Functions

Formulas

f'(x) = 3x^2 - 6x

Theorems

First Derivative Test

Suitable Grade Level

Grades 11-12