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Find the critical points of the function f(x)=x3−3x+1f(x) = x^3 - 3x + 1.
Apr 23, 2024
Find the critical points of the function f(x)=x3−3x+1f(x) = x^3 - 3x + 1.
Generated Graph
Solution by Steps
step 1
To find the critical points of the function f(x)=x3−3x+1f(x) = x^3 - 3x + 1, we need to find the values of xx where the first derivative f′(x)f'(x) is equal to zero or undefined
step 2
First, we calculate the derivative of f(x)f(x) with respect to xx: f′(x)=3x2−3f'(x) = 3x^2 - 3
step 3
Set the derivative equal to zero to find critical points: 3x2−3=03x^2 - 3 = 0
step 4
Factor out the common factor of 3: 3(x2−1)=03(x^2 - 1) = 0
step 5
Apply the zero product property: x2−1=0x^2 - 1 = 0
step 6
Solve the equation x2−1=0x^2 - 1 = 0 by factoring it as (x+1)(x−1)=0(x + 1)(x - 1) = 0
step 7
Set each factor equal to zero and solve for xx: x+1=0x + 1 = 0 or x−1=0x - 1 = 0
step 8
The solutions are x=−1x = -1 and x=1x = 1. These are the critical points of the function
Answer
The critical points of the function f(x)=x3−3x+1f(x) = x^3 - 3x + 1 are x=−1x = -1 and x=1x = 1.
Key Concept
Critical Points of a Function
Explanation
Critical points occur where the derivative of the function is zero or undefined. In this case, setting the first derivative equal to zero and solving for xx gives the critical points.
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