Question

Math

Posted 4 months ago

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

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Answer from Sia

Posted 4 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 $f(x)$ with respect to $x$: $f'(x) = 3x^2 - 3$

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

Apply the zero product property: $x^2 - 1 = 0$

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 and solve for $x$: $x + 1 = 0$ or $x - 1 = 0$

step 8

The solutions are $x = -1$ and $x = 1$. These are the critical points of the function

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 occur where the derivative of the function is zero or undefined. In this case, setting the first derivative equal to zero and solving for $x$ gives the critical points.

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