Question

Math

Posted 3 months ago

`Determine the critical points of the function $f(x) = x^4 - 4x^2 + 3$.`

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

To find the critical points of the function $f(x) = x^4 - 4x^2 + 3$, we need to find the values of $x$ for which the first derivative $f'(x)$ is equal to zero

step 2

We calculate the first derivative of $f(x)$: $f'(x) = \frac{d}{dx}(x^4 - 4x^2 + 3)$

step 3

Simplifying the derivative, we get $f'(x) = 4x^3 - 8x$

step 4

Set the derivative equal to zero: $4x^3 - 8x = 0$

step 5

Factor out the common term $4x$: $4x(x^2 - 2) = 0$

step 6

Set each factor equal to zero: $x = 0$ and $x^2 - 2 = 0$

step 7

Solve for $x$ in $x^2 - 2 = 0$: $x = \pm\sqrt{2}$

step 8

The critical points are $x = 0$ and $x = \pm\sqrt{2}$

Answer

The critical points of the function $f(x) = x^4 - 4x^2 + 3$ are $x = 0$ and $x = \pm\sqrt{2}$.

Key Concept

Critical Points of a Function

Explanation

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

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