How do you find the maxima and minima of the function #f(x)=x^3+3x^224x+3#?
maxima at
#(4,83)#
minima at#(2,25)#
We have:
So the maxima and minima are:
We can confirm these results graphically: graph{x^3+3x^224x+3 [10, 10, 50, 100.]}
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To find the maxima and minima of the function ( f(x) = x^3 + 3x^2  24x + 3 ), you need to follow these steps:
 Find the derivative of the function, ( f'(x) ).
 Set the derivative equal to zero and solve for ( x ) to find critical points.
 Use the second derivative test or the first derivative test to determine whether these critical points are maxima or minima.
Let's go through these steps:

Find the derivative of ( f(x) ): [ f'(x) = 3x^2 + 6x  24 ]

Set the derivative equal to zero and solve for ( x ): [ 3x^2 + 6x  24 = 0 ] [ x^2 + 2x  8 = 0 ] [ (x + 4)(x  2) = 0 ]
Solving for ( x ): [ x + 4 = 0 \implies x = 4 ] [ x  2 = 0 \implies x = 2 ]
So, the critical points are ( x = 4 ) and ( x = 2 ).
 Use the first derivative test or the second derivative test to determine whether these critical points are maxima or minima.
First Derivative Test:
 For ( x = 4 ): ( f'(4) = 3(4)^2 + 6(4)  24 = 48  24  24 = 0 ) (changing from increasing to decreasing), indicating a local maximum.
 For ( x = 2 ): ( f'(2) = 3(2)^2 + 6(2)  24 = 12 + 12  24 = 0 ) (changing from decreasing to increasing), indicating a local minimum.
Therefore, the function has a local maximum at ( x = 4 ) and a local minimum at ( x = 2 ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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