Question

Math

Posted about 2 months ago

Great answer!

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

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Answer from Sia

Posted about 2 months ago

Solution by Steps

step 1

To determine the critical points of the function $f(x) = x^4 - 4x^2 + 3$, we first need to find its derivative

step 2

The derivative of $f(x)$ is $f'(x) = \frac{d}{dx}(x^4 - 4x^2 + 3)$

step 3

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

step 4

To find the critical points, we set the derivative equal to zero: $4x^3 - 8x = 0$

step 5

Factor the equation: $4x(x^2 - 2) = 0$

step 6

Solve for $x$: $4x = 0$ or $x^2 - 2 = 0$

step 7

This gives us $x = 0$ or $x = \pm \sqrt{2}$

Answer

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

Key Concept

Critical Points

Explanation

Critical points of a function occur where its derivative is zero or undefined. In this case, we found the derivative, set it to zero, and solved for $x$ to find the critical points.

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