How do you find the maximum and minimum values of the function #f(x)= x - ((64x)/(x+4))# on the interval [0,13]?

Answer 1

There is only a local minimum at #(12, -36)#

Calculate the first derivative of #f(x)#

The derivative of

#(x)'=1#

And by the quotient rule

#((64x)/(x+4))'=(64(x+4)-1(64x))/(x+4)^2#
#=256/(x+4)^2#

Therefore,

#f'(x)=1-256/(x+4)^2=(((x+4)^2)-64)/(x+4)^2#
#=(x^2+8x+16-256)/(x+4)^2#
#=(x^2+8x-240)/(x+4)^2#
#=((x+20)(x-12))/(x+4)^2#

The critical points are when

#f'(x)=0#
#=>#, #{(x=-20),(x=12):}#
Build a variation chart in the interval #[0,13]#
#color(white)(aaaa)##"Interval"##color(white)(aaaa)##(0,12)##color(white)(aaaa)##(12,13)#
#color(white)(aaaa)##"sign f'(x)"##color(white)(aaaaa)##-##color(white)(aaaaaaa)##+#
#color(white)(aaaa)##" f(x)"##color(white)(aaaaaaaa)##↘##color(white)(aaaaaaa)##↗#
There is only a local minimum at #(12, -36)#

graph{x-((64x)/(x+4)) [-73.8, 113.7, -63, 30.8]}

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Answer 2

To find the maximum and minimum values of the function ( f(x) = x - \frac{64x}{x+4} ) on the interval ([0,13]), follow these steps:

  1. Find the critical points of the function within the given interval by setting its derivative equal to zero and solving for ( x ).

  2. Evaluate the function at the critical points as well as at the endpoints of the interval ([0,13]).

  3. The maximum and minimum values will be the maximum and minimum function values obtained from the critical points and endpoints.

  4. Check for critical points that lie outside the interval, as they won't be considered in this case.

After evaluating the function, you can determine the maximum and minimum values within the given interval.

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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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