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Question
Math
Posted 3 months ago
Determine the critical points of the function f(x)=x44x2+3f(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)=x44x2+3 f(x) = x^4 - 4x^2 + 3 , we need to find the values of x x for which the first derivative f(x) f'(x) is equal to zero
step 2
We calculate the first derivative of f(x) f(x) : f(x)=ddx(x44x2+3) f'(x) = \frac{d}{dx}(x^4 - 4x^2 + 3)
step 3
Simplifying the derivative, we get f(x)=4x38x f'(x) = 4x^3 - 8x
step 4
Set the derivative equal to zero: 4x38x=0 4x^3 - 8x = 0
step 5
Factor out the common term 4x 4x : 4x(x22)=0 4x(x^2 - 2) = 0
step 6
Set each factor equal to zero: x=0 x = 0 and x22=0 x^2 - 2 = 0
step 7
Solve for x x in x22=0 x^2 - 2 = 0 : x=±2 x = \pm\sqrt{2}
step 8
The critical points are x=0 x = 0 and x=±2 x = \pm\sqrt{2}
Answer
The critical points of the function f(x)=x44x2+3 f(x) = x^4 - 4x^2 + 3 are x=0 x = 0 and x=±2 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 x .

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