Question

Math

Posted 5 months ago

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

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

To find the critical points of the function $f(x) = x^3 - 3x + 1$, we first need to find its derivative

step 2

The derivative of $f(x)$ is $f'(x) = 3x^2 - 3$

step 3

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

step 4

Solve the equation $3x^2 - 3 = 0$ for $x$

step 5

Simplify the equation: $3(x^2 - 1) = 0$

step 6

Factor the equation: $3(x - 1)(x + 1) = 0$

step 7

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

step 8

Solve for $x$: $x = 1$ 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

Explanation

Critical points of a function occur where its derivative is zero or undefined. In this case, we found the points where the derivative $f'(x) = 3x^2 - 3$ is zero.

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