Question

Math

Posted 6 months ago

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`Find the critical points of the function $f(x) = x^3 - 3x + 1$.`

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

To find the critical points of the function $f(x) = x^3 - 3x + 1$, we need to find the values of $x$ where the first derivative $f'(x)$ is equal to zero or undefined

step 2

First, we calculate the derivative of $f(x)$. The derivative of $x^3$ is $3x^2$, the derivative of $-3x$ is $-3$, and the derivative of a constant is 0. So, $f'(x) = 3x^2 - 3$

step 3

Set the derivative equal to zero to find critical points: $3x^2 - 3 = 0$

step 4

Factor out the common factor of 3: $3(x^2 - 1) = 0$

step 5

Set the factored expression equal to zero: $x^2 - 1 = 0$

step 6

Solve the equation $x^2 - 1 = 0$ by factoring it as $(x - 1)(x + 1) = 0$

step 7

Set each factor equal to zero: $x - 1 = 0$ and $x + 1 = 0$

step 8

Solve for $x$ in each equation: $x = 1$ and $x = -1$

Answer

The critical points of the function $f(x) = x^3 - 3x + 1$ are $x = 1$ and $x = -1$.

Key Concept

Critical Points of a Function

Explanation

Critical points of a function occur where its derivative is zero or undefined. In this case, the critical points are found by setting the first derivative of the function equal to zero and solving for $x$.

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