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Find the critical points of the function f(x)=x3−3x+1f(x) = x^3 - 3x + 1.
Jan 6, 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+1 f(x) = x^3 - 3x + 1 , we need to find the values of x x 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) . The derivative of x3 x^3 is 3x2 3x^2 , the derivative of −3x -3x is −3 -3 , and the derivative of a constant is 0. So, f′(x)=3x2−3 f'(x) = 3x^2 - 3
step 3
Set the derivative equal to zero to find the critical points: 3x2−3=0 3x^2 - 3 = 0
step 4
Factor out the common factor of 3: 3(x2−1)=0 3(x^2 - 1) = 0
step 5
Set the factor x2−1 x^2 - 1 equal to zero: x2−1=0 x^2 - 1 = 0
step 6
Solve the equation x2−1=0 x^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 x x : x+1=0 x + 1 = 0 gives x=−1 x = -1 , and x−1=0 x - 1 = 0 gives x=1 x = 1
Answer
The critical points of the function f(x)=x3−3x+1 f(x) = x^3 - 3x + 1 are x=−1 x = -1 and x=1 x = 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, the critical points are found by setting the first derivative equal to zero and solving for x x .
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