Question: Determine the critical points of f(x) = x^3 - 3x + 2.
Options:
-1, 1
0, 2
1, -2
2, -1
Correct Answer: -1, 1
Solution:
Setting f\'(x) = 3x^2 - 3 = 0 gives x^2 = 1, so critical points are x = -1 and x = 1.
Determine the critical points of f(x) = x^3 - 3x + 2.
Practice Questions
Q1
Determine the critical points of f(x) = x^3 - 3x + 2.
-1, 1
0, 2
1, -2
2, -1
Questions & Step-by-Step Solutions
Determine the critical points of f(x) = x^3 - 3x + 2.
Correct Answer: x = -1 and x = 1
Step 1: Start with the function f(x) = x^3 - 3x + 2.
Step 2: Find the derivative of the function, which is f'(x). The derivative of f(x) is f'(x) = 3x^2 - 3.
Step 3: Set the derivative equal to zero to find critical points: 3x^2 - 3 = 0.
Step 4: Simplify the equation: 3x^2 = 3.
Step 5: Divide both sides by 3: x^2 = 1.
Step 6: Solve for x by taking the square root of both sides: x = ±1.
Step 7: The critical points are x = -1 and x = 1.
Finding Critical Points – This involves taking the derivative of a function and setting it to zero to find points where the function's slope is zero.
Derivative Calculation – Understanding how to differentiate polynomial functions correctly.
Identifying Local Extrema – Recognizing that critical points can indicate local maxima, minima, or points of inflection.
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