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Question
Math
Posted about 2 months ago
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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 about 2 months ago
Solution by Steps
step 1
To determine the critical points of the function f(x)=x44x2+3f(x) = x^4 - 4x^2 + 3, we first need to find its derivative
step 2
The derivative of f(x)f(x) is f(x)=ddx(x44x2+3)f'(x) = \frac{d}{dx}(x^4 - 4x^2 + 3)
step 3
Calculating the derivative, we get f(x)=4x38xf'(x) = 4x^3 - 8x
step 4
To find the critical points, we set the derivative equal to zero: 4x38x=04x^3 - 8x = 0
step 5
Factor the equation: 4x(x22)=04x(x^2 - 2) = 0
step 6
Solve for xx: 4x=04x = 0 or x22=0x^2 - 2 = 0
step 7
This gives us x=0x = 0 or x=±2x = \pm \sqrt{2}
Answer
The critical points of the function f(x)=x44x2+3f(x) = x^4 - 4x^2 + 3 are x=0x = 0, x=2x = \sqrt{2}, and x=2x = -\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 xx to find the critical points.

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