`f(x) = 5 + 54x - 2x^3, [0, 4]` Find the absolute maximum and minimum values of f on the given interval

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Given: `f(x)=5+54x-2x^3,[0,4]`

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

`f'(x)=54-6x^2=0`

`54=6x^2`

`9=x^2`

`x=3,x=-3`

The critical value is x=3. The value x=-3 is not in the given interval [0, 4]. Therefore x=-3 will not be used to find the absolute maximum...

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Given: `f(x)=5+54x-2x^3,[0,4]`

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

`f'(x)=54-6x^2=0`

`54=6x^2`

`9=x^2`

`x=3,x=-3`

The critical value is x=3. The value x=-3 is not in the given interval [0, 4]. Therefore x=-3 will not be used to find the absolute maximum or absolute minimum value. Plug in the critical value x=3 and the endpoints of the interval [0, 4] into the original f(x) function. 

f(0)=5

f(3)=113

f(4)=93

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

The absolute maximum occurs at the point (3, 113).

The absolute minimum occurs at the point (0, 5).

Approved by eNotes Editorial Team