Maxima And Minima

Concept: Calculus - Application of Derivatives (Maxima and Minima on a Closed Interval).

Step 1: Solve the inequality to find the domain set . The given set condition is . Rearrange into a standard quadratic inequality: . Factor the quadratic: . Using the wavy curve method, the solution is the closed interval . Thus, we only evaluate the function for .

Step 2: Find the first derivative of the function. Differentiate with respect to . . Factor out the common term 9: . Further factor the quadratic expression: .

Step 3: Analyze the sign of the derivative on the specified interval. We need to check the behavior of strictly within our domain . For any value of between 5 and 6, is positive and is positive. Therefore, for all .

Step 4: Determine the monotonicity to find the maximum point. Because the first derivative is strictly greater than 0 on the entire interval , the function is strictly increasing over this domain. For a strictly increasing function, the maximum value will always occur at the rightmost endpoint of the interval.

Step 5: Calculate the maximum value at the endpoint. Substitute the upper bound into the original function: . . . $ $

We have, Maximize Subject to constraints The graph of inequality are

Maximum value