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Determine the critical points of the function

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

Nov 23, 2023

Determine the critical points of the function

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

Here is a Step-by-Step solution: #step 1#

Find the derivative of the function

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

#step 2#

Calculate

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

#step 3#

Set the derivative equal to zero to find critical points:

4x^3 - 8x = 0

#step 4#

Factor out the common term:

4x(x^2 - 2) = 0

#step 5#

Solve for

x

x = 0, x = \sqrt{2}, x = -\sqrt{2}

Answer

The critical points of the function are

x = 0, x = \sqrt{2}, x = -\sqrt{2}

Key Concept

Critical points occur where the derivative is zero or undefined.

Explanation

The critical points found are the values of

x

where the first derivative of the function equals zero, indicating potential local maxima, minima, or points of inflection.