Step 1: Define the function and interval

We are given the function $f(x) = \cos x + \sin^2 x$ on the closed interval $[0, \pi]$. Using the identity $\sin^2 x = 1 - \cos^2 x$, we can rewrite the function in terms of $\cos x$:

Step 2: Find the critical points

Let $u = \cos x$. Since $x \in [0, \pi]$, $u \in [-1, 1]$. The function becomes $g(u) = -u^2 + u + 1$. To find the critical points, we differentiate with respect to $u$:

Setting $g'(u) = 0$, we get $u = 1/2$. Since $1/2 \in [-1, 1]$, this is a valid critical point.

Step 3: Evaluate at critical points and boundaries

We evaluate $g(u)$ at $u = -1, 1/2, 1$:

At $u = -1$ (which corresponds to $x = \pi$): $g(-1) = -(-1)^2 + (-1) + 1 = -1 - 1 + 1 = -1$.

At $u = 1$ (which corresponds to $x = 0$): $g(1) = -(1)^2 + (1) + 1 = 1$.

At $u = 1/2$ (which corresponds to $x = \pi/3$): $g(1/2) = -(1/2)^2 + (1/2) + 1 = -1/4 + 1/2 + 1 = 5/4$.