Given: `f(x)=12+4x-x^2,[0,5]`

Find the critical value(s) of the function by setting the first derivative equal to zero and solving for the x value(s).

`f'(x)=4-2x=0`

`4=2x`

`2=x`

The value x=2 is a critical number. Plug in the x=2 and the endpoints of the interval into the original f(x) function.

f(0)=12

f(2)=16

f(5)=7

...

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Given: `f(x)=12+4x-x^2,[0,5]`

Find the critical value(s) of the function by setting the first derivative equal to zero and solving for the x value(s).

`f'(x)=4-2x=0`

`4=2x`

`2=x`

The value x=2 is a critical number. Plug in the x=2 and the endpoints of the interval into the original f(x) function.

f(0)=12

f(2)=16

f(5)=7

Examine the f(x) values to determine the absolute maximum and absolute minimum.

The absolute maximum occurs at the point (2, 16).

The absolute minimum occurs at the point (5, 7).