Math Problem Statement

Given the function f(x) = x^3 - 3x + 2, find the value of x where the local minimum occurs.

Solution

The problem asks to find the value of xx where the local minimum of the function f(x)=x33x+2f(x) = x^3 - 3x + 2 occurs.

Steps to solve:

  1. Find the derivative of f(x)f(x): The derivative f(x)f'(x) helps identify critical points where the slope of the function is zero (possible local minima or maxima). f(x)=ddx(x33x+2)=3x23f'(x) = \frac{d}{dx}(x^3 - 3x + 2) = 3x^2 - 3

  2. Set the derivative equal to zero: To find critical points: 3x23=03x^2 - 3 = 0 Simplify: x2=1x=±1x^2 = 1 \quad \Rightarrow \quad x = \pm 1

  3. Determine the nature of the critical points: Use the second derivative f(x)f''(x) to check whether each critical point is a local minimum or maximum.

    The second derivative is: f(x)=ddx(3x23)=6xf''(x) = \frac{d}{dx}(3x^2 - 3) = 6x

    • For x=1x = 1: f(1)=6(1)=6(positive, so local minimum at x=1)f''(1) = 6(1) = 6 \quad (\text{positive, so local minimum at } x = 1)
    • For x=1x = -1: f(1)=6(1)=6(negative, so local maximum at x=1)f''(-1) = 6(-1) = -6 \quad (\text{negative, so local maximum at } x = -1)

Thus, the local minimum occurs at x=1x = 1.

The correct response is Response 2: x=1x = 1.


Would you like further details on any steps or concepts involved in this solution?

Here are five related questions to deepen your understanding:

  1. How do second derivatives help in identifying local minima and maxima?
  2. What are the necessary conditions for a critical point to be a saddle point?
  3. Can a cubic function have more than one local minimum?
  4. How does concavity relate to the second derivative of a function?
  5. How would the solution change if the function had higher-order terms?

Tip: Always check critical points using the second derivative test to confirm whether they are minima, maxima, or saddle points.

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Math Problem Analysis

Mathematical Concepts

Calculus
Critical Points
Derivatives
Second Derivative Test

Formulas

f'(x) = 3x^2 - 3 (First Derivative)
f''(x) = 6x (Second Derivative)
Local minimum condition: f''(x) > 0

Theorems

First Derivative Test
Second Derivative Test

Suitable Grade Level

Grades 10-12