2. Find all critical values, local extrema and the absolute maximum and absolute minimum values of f(x)= x(4-x)2 on the interval [0,5].

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2. Find all critical values, local extrema and the absolute maximum and absolute minimum values of f(x)= x(4-x)on the interval [0,5].

 

Expert Solution
Step 1

For maxima, the value of f’’(x) at the root of f’(x) must be less than 0.

For minima, the value of f’’(x) at the root of f’(x) must be greater than 0.

Step 2

 Simplify the function.

f(x)=x(4-x)2

=x(16+x2-8x)

=16x+x3-8x2

Step 3

Find the derivative of the function.

f’(x)=16+3x2-16x

Equate the function to 0 and determine the root of f’(x).

Calculus homework question answer, step 3, image 1

Thus, the critical values are x= 4/3 , 4

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