How do you find the exact relative maximum and minimum of the function of #f(x) = 4x+6x^1#?
Find the critical numbers. Test the critical numbers. Evaluate
This is a good example to use the second derivative test for.
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To find the exact relative maximum and minimum of the function ( f(x) = 4x + 6x^{1} ), follow these steps:
 Find the derivative of the function ( f'(x) ).
 Set ( f'(x) ) equal to zero and solve for ( x ) to find critical points.
 Use the second derivative test or analyze the behavior of ( f'(x) ) around the critical points to determine the nature of the extremum (maximum or minimum).
Let's go through these steps:

Find the derivative of the function ( f(x) ): [ f'(x) = 4  6x^{2} ]

Set ( f'(x) ) equal to zero and solve for ( x ) to find critical points: [ 4  6x^{2} = 0 ] [ 6x^{2} = 4 ] [ x^{2} = \frac{4}{6} = \frac{2}{3} ] [ x^2 = \frac{3}{2} ] [ x = \pm \sqrt{\frac{3}{2}} = \pm \frac{\sqrt{6}}{2} ]
So, the critical points are ( x = \frac{\sqrt{6}}{2} ) and ( x = \frac{\sqrt{6}}{2} ).
 Determine the nature of the extremum using the second derivative test: [ f''(x) = 12x^{3} ]
Evaluate ( f''(x) ) at the critical points:
 At ( x = \frac{\sqrt{6}}{2} ), ( f''\left(\frac{\sqrt{6}}{2}\right) = 12\left(\frac{\sqrt{6}}{2}\right)^{3} = 12\sqrt{6} ), which is positive. So, there is a relative minimum at ( x = \frac{\sqrt{6}}{2} ).
 At ( x = \frac{\sqrt{6}}{2} ), ( f''\left(\frac{\sqrt{6}}{2}\right) = 12\left(\frac{\sqrt{6}}{2}\right)^{3} = 12\sqrt{6} ), which is negative. So, there is a relative maximum at ( x = \frac{\sqrt{6}}{2} ).
Therefore, the exact relative minimum occurs at ( x = \frac{\sqrt{6}}{2} ) and the exact relative maximum occurs at ( x = \frac{\sqrt{6}}{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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