What are the local extrema, if any, of #f(x)= 120x^5 - 200x^3#?
Local maximum of
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To find the local extrema of ( f(x) = 120x^5 - 200x^3 ), we first need to find its critical points. Critical points occur where the derivative of the function is zero or undefined.
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Find the derivative of the function: [ f'(x) = 600x^4 - 600x^2 ]
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Set the derivative equal to zero and solve for ( x ): [ 600x^4 - 600x^2 = 0 ] [ 600x^2(x^2 - 1) = 0 ] [ x = 0, \pm 1 ]
These are the critical points.
- Determine the nature of the critical points by using the first derivative test:
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When ( x = 0 ): Substitute ( x = 0 ) into the second derivative ( f''(x) ). [ f''(0) = 0 ]
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When ( x = 1 ): Substitute ( x = 1 ) into the second derivative ( f''(x) ). [ f''(1) = 1200 ]
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When ( x = -1 ): Substitute ( x = -1 ) into the second derivative ( f''(x) ). [ f''(-1) = 1200 ]
Since ( f''(1) ) and ( f''(-1) ) are both positive, and ( f''(0) = 0 ), there are local minima at ( x = -1 ) and ( x = 1 ), and ( x = 0 ) is a point of inflection.
So, the local extrema of ( f(x) = 120x^5 - 200x^3 ) are:
- Local minimum at ( x = -1 )
- Local minimum at ( x = 1 )
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To find the local extrema of ( f(x) = 120x^5 - 200x^3 ):
-
Find the derivative of the function. [ f'(x) = 600x^4 - 600x^2 ]
-
Set the derivative equal to zero and solve for ( x ) to find critical points. [ 600x^4 - 600x^2 = 0 ] [ 600x^2(x^2 - 1) = 0 ] [ x = 0, \pm 1 ]
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Test the critical points in the second derivative to determine the nature of the extrema. [ f''(x) = 2400x^3 - 1200x ]
For ( x = 0 ): [ f''(0) = 0 ]
For ( x = 1 ): [ f''(1) = 2400(1)^3 - 1200(1) = 2400 - 1200 = 1200 > 0 ]
For ( x = -1 ): [ f''(-1) = 2400(-1)^3 - 1200(-1) = -2400 - (-1200) = -1200 < 0 ]
Therefore, at ( x = 1 ), there is a local minimum, and at ( x = -1 ), there is a local maximum.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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