Part b
Find the value of \(x\) that maximizes the area by setting the derivative of \(A(x)\) equal to zero and solving for \(x\).
Determining Absolute Extrema of a Function
Consider the function \(f(x)= x^3 – 3x^2 – 9x + 27\) defined on the closed interval \([0, 5]\). (a) Find the critical point(s) of \(f(x)\) in the interval. (b) Determine the absolute maximum and minimum values of \(f(x)\) on \([0, 5]\) by comparing the values at the critical point(s) and at the endpoints.
Part b
Differentiate the area function \(A(x) = x(50 – x)\) with respect to \(x\) to find \(\frac{dA}{dx}\), and solve \(\frac{dA}{dx} = 0\) to find the value of \(x\) that maximizes the area.
Part c
Determine the value of \(x\) that maximizes \(f(x)\) by setting \(f'(x) = 0\), and verify the result using the second derivative test.