How do you find local maximum value of f using the first and second derivative tests: #f(x)= x^2 + 8x 12#?
To determine whether or not the point is a local maximum, you could use either the first or second derivative tests.
For the first derivative test , create a sign chart where the important values are the critical numbers. If the signs change from positive to negative, the point is a local maximum.
For the second derivative test , plug the critical number(s) into the second derivative. If the value is negative, the function is concave down at that point meaning the point is a local maximum.
Now, let's do the work:
First derivative test:
We can prove the same thing with the...
Second derivative test:
Thus, the second derivative is ALWAYS positive, and the function is always concave up, which results in a local minimum.
Therefore, the function has no local maxima.
We can check a graph, even though it is obvious that the graph will form a parabola facing up, which will have only a minimum at its vertex: graph{x^2+8x12 [65, 66.67, 36.6, 29.23]}
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To find the local maximum value of ( f(x) = x^2 + 8x  12 ) using the first and second derivative tests:

Find the first derivative of ( f(x) ) to locate critical points. [ f'(x) = 2x + 8 ]

Set ( f'(x) = 0 ) and solve for ( x ) to find critical points. [ 2x + 8 = 0 ] [ x = 4 ]

Find the second derivative of ( f(x) ) to determine concavity. [ f''(x) = 2 ]

Test the concavity of ( f(x) ) at the critical point ( x = 4 ):
 Since ( f''(4) = 2 > 0 ), the function is concave up at ( x = 4 ).

Use the first derivative test to confirm if ( x = 4 ) is a local maximum:
 Since ( f'(4) = 2(4) + 8 = 0 ) and the function changes from increasing to decreasing at ( x = 4 ), it indicates a local maximum.

To find the local maximum value, substitute ( x = 4 ) into the original function: [ f(4) = (4)^2 + 8(4)  12 = 16  32  12 = 28 ]
Thus, the local maximum value of ( f(x) ) is ( 28 ) at ( x = 4 ).
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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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