How do you find the extrema and points of inflection for #f(x) =(3x)/((x+8)^2)#?

Answer 1

Once derivative:

#((3x)/(x+8)^2)' = (3(x+8)^2-3x(2(x+8)))/(x+8)^4 = -(3(x-8))/(x+8)^3#
Root is #x=8# We have global maxima at #x = 8# and #y = 3/32#
Derivate again #-(3(x-8))/(x+8)^3# = #-(3(x+8)^3-9(x-8)(x+8)^2)/(x+8)^6 = (6(x-16))/(x+8)^4#
Root is #x = 16# We have a points of inflection at #x = 16# and #y = 1/12#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the extrema and points of inflection for ( f(x) = \frac{3x}{(x+8)^2} ), follow these steps:

  1. 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} )

  2. Simplify ( f'(x) ) to find critical points.

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

  3. Find the second derivative, ( f''(x) ), of the function.

    ( f''(x) ) can be found by differentiating ( f'(x) ) obtained in step 1.

  4. 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.
  5. 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.

  6. 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.
  7. 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} ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7