Find the maximum value of f(x) = $2x^3 - 9x^2 + 12x + 5$ on the interval [0, 3].

Step 1: Find the first derivative

To find the critical points of the function, we first need to calculate its derivative, f'(x). We apply the power rule for differentiation to each term in the function f(x).

Step 2: Find critical points

Critical points occur where the first derivative f'(x) is equal to zero or undefined. We set f'(x) = 0 and solve the resulting quadratic equation for x. Factoring the quadratic equation gives us the critical points $x=1$ and $x=2$. Both of these points lie within the given interval [0, 3].

Step 3: Evaluate function at critical points and endpoints

To find the maximum value of the function on the closed interval [0, 3], we must evaluate the function at the critical points that lie within the interval, as well as at the endpoints of the interval. We calculate f(0), f(1) $, f(2)$, and f(3).

Step 4: Determine the maximum value

By comparing the values of the function at the critical points and the endpoints, we can identify the maximum value. The values are 5, 10, 9, 14. The largest among these is 14.

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