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

Problem

ƒ(x)=3*x2−1,[1,3]

Solution

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

d(ƒ(x))/d(x)=6*x

2. Identify critical points by setting the derivative equal to zero and solving for x

6*x=0

x=0

3. Check the interval to see if the critical point x=0 falls within [1,3] Since 0 is not in the interval, we only need to evaluate the function at the endpoints.

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

ƒ(1)=3*(1)2−1

ƒ(1)=2

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

ƒ(3)=3*(3)2−1

ƒ(3)=26

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

Final Answer

Absolute Max=26,Absolute Min=2

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