Find the Absolute Max and Min over the Interval f(x)=6x^2 , [1,3]

Problem

ƒ(x)=6*x2,[1,3]

Solution

1. Find the derivative of the function to locate any critical points within the interval.

d(ƒ(x))/d(x)=12*x

2. Set the derivative to zero to solve for critical values.

12*x=0

x=0

3. Check the critical point to see if it lies within the closed interval [1,3]

0∉[1,3]

Since the critical point is outside the interval, we only need to evaluate the function at the endpoints.

4. Evaluate the function at the left endpoint x=1

ƒ(1)=6*(1)2

ƒ(1)=6

5. Evaluate the function at the right endpoint x=3

ƒ(3)=6*(3)2

ƒ(3)=54

6. Compare the values to determine the absolute maximum and minimum.
   The smallest value is 6 and the largest value is 54

Final Answer

Absolute Max=54,Absolute Min=6

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