How do you find the extrema and points of inflection for #f(x) =(3x)/((x+8)^2)#?
Once derivative:
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To find the extrema and points of inflection for ( f(x) = \frac{3x}{(x+8)^2} ), follow these steps:

Find the first derivative, ( f'(x) ), of the function.
( f'(x) ) can be found using the quotient rule:
( f'(x) = \frac{(3)(x+8)^2  3x(2)(x+8)}{(x+8)^4} )

Simplify ( f'(x) ) to find critical points.
Set ( f'(x) = 0 ) and solve for ( x ) to find critical points.

Find the second derivative, ( f''(x) ), of the function.
( f''(x) ) can be found by differentiating ( f'(x) ) obtained in step 1.

Determine the nature of critical points using the second derivative test.
Evaluate ( f''(x) ) at each critical point:
 If ( f''(x) > 0 ), the point is a local minimum.
 If ( f''(x) < 0 ), the point is a local maximum.
 If ( f''(x) = 0 ), the test is inconclusive.

Find points of inflection by setting ( f''(x) = 0 ) and solving for ( x ).
Points where ( f''(x) = 0 ) or changes sign are potential points of inflection.

Verify points of inflection by analyzing the concavity of the function around these points.
Use the sign of ( f''(x) ) to determine concavity:
 If ( f''(x) > 0 ), the function is concave up.
 If ( f''(x) < 0 ), the function is concave down.

Check for any horizontal or vertical asymptotes, if applicable.
Analyze the behavior of the function as ( x ) approaches positive or negative infinity.
Perform these steps to determine the extrema and points of inflection for ( f(x) = \frac{3x}{(x+8)^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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