On the segment [-1; 1], the function y = cos x is convex, concave, or has an inflection point.
May 28, 2021 | education
| The second derivative is responsible for the convexity (concavity) and inflection points, we find it:
y ‘= (cos (x))’ = -sin (x).
y ” = (-sin (x)) ‘= -cos (x).
We equate it to zero:
-cos (x) = 0.
The roots of an equation of the form cos (x) = a are determined by the formula:
x = arccos (a) + – 2 * π * n, where n is a natural number.
x = arccos (0) + – 2 * π * n.
x = π / 2 + – 2 * π * n.
There is no x belonging to the given interval, we find the value of the second derivative at the point x0 = 0.
y ” = -cos (0) = -π / 2 <0 hence the function is convex.
Answer: the function is convex.
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