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Determine the critical points of the function

f(x) = x^4 - 4x^2 + 3

Mar 16, 2024

Determine the critical points of the function

f(x) = x^4 - 4x^2 + 3

Generated Graph

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

Applying the power rule for differentiation, we get

f'(x) = 4x^3 - 8x

step 4

Set the first 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:

4x = 0

and

x^2 - 2 = 0

step 7

Solve for

x

: From

4x = 0

, we get

x = 0

. From

x^2 - 2 = 0

, we get

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

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 equal to zero and solving for

x