Finding Absolute Minima and Maxima of f(x) = sin(x) + cos^2(x) on [0, π]

Math Problem Statement

Let f(x) = \sin(x)+\cos^2(x). Find all points of absolute minima and maxima on the interval [0,\pi]. As your answer, input the sum of the values of f at all points the absolute minima and maxima occur in the given interval.

Solution

To find the points of absolute minima and maxima of the function $f(x) = \sin(x) + \cos^2(x)$ on the interval $[0, \pi]$ , we will follow these steps:

Find the derivative of $f(x)$ .
Determine the critical points by setting the derivative equal to zero.
Evaluate $f(x)$ at the critical points and at the endpoints of the interval $[0, \pi]$ .
Identify the absolute

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Calculus
Critical Points

Formulas

f'(x) = \cos(x) - 2\cos(x)\sin(x)
f(x) = \sin(x) + \cos^2(x)

Theorems

First Derivative Test
Critical Point Theorem

Suitable Grade Level

Grades 11-12 or Early College

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