How would you find the inflection point and the concavity of #g(x) = (5x - 2.6) / (5x - 6.76)^2#? I know I have to take the 2nd derivative but i'm not sure how because of the odd way this function is set up.?

Answer 1
Use the quotient rule to find #g'(x)#, then simplify if possible, then use the quotient rule again to find #g''(x)#.
#g(x)= (5x-2.6)/(5x-6.76)^2#
#g'(x)=( 5(5x-6.76)^2 - (5x-2.6)2(5x-6.76)(5))/(5x-6.76)^4#
# = ( 5(5x-6.76)[(5x-6.76) - 2(5x-2.6)])/(5x-6.76)^4#
# = ( 5[5x-6.76 - 10x + 5.2])/(5x-6.76)^3#
#g'(x) = ( -5(5x + 1.56))/(5x-6.76)^3#
Differentiate again to get #g''(x)#.
After finding #g''(x)#, proceed as in any other question about concavity.
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Answer 2

To find the inflection point and the concavity of ( g(x) = \frac{{5x - 2.6}}{{(5x - 6.76)^2}} ), you can follow these steps:

  1. Find the first derivative of ( g(x) ).
  2. Find the second derivative of ( g(x) ).
  3. Set the second derivative equal to zero and solve for ( x ) to find potential inflection points.
  4. Use the second derivative test to determine the concavity at these points.

Let's start by finding the first and second derivatives of ( g(x) ):

  1. ( g'(x) = \frac{{d}}{{dx}} \left( \frac{{5x - 2.6}}{{(5x - 6.76)^2}} \right) )
  2. ( g''(x) = \frac{{d^2}}{{dx^2}} \left( \frac{{5x - 2.6}}{{(5x - 6.76)^2}} \right) )

After finding the second derivative, you can proceed to find the potential inflection points and determine the concavity using the second derivative test.

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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.

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