ow many inflection points does the function h(x) = x+cos(x) have in the interval [0, 2π]?A. 0B. 1C. 2D. 3E. 4
Question
Solution
To find the inflection points of the function h(x) = x + cos(x), we first need to find the second derivative of the function.
The first derivative of the function h(x) is h'(x) = 1 - sin(x).
The second derivative of the function h(x) is h''(x) = -cos(x).
An inflection point occurs where the second derivative changes sign.
The function cos(x) is positive in the intervals [0, π/2] and [3π/2, 2π] and negative in the interval [π/2, 3π/2] within the interval [0, 2π].
Therefore, the second derivative changes sign at x = π/2 and x = 3π/2.
So, the function h(x) = x + cos(x) has 2 inflection points in the interval [0, 2π].
The answer is C. 2.
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