First slide

Monotonicity

Question

On which of the following intervals, the function $x^{200} + \sin x - 1$ is strictly increasing?

Difficult

A

(–1, 1)
B

(0, 1)
C

$(\frac{π}{2}, π)$
D

$(0, \frac{π}{2})$

Solution

$f' (x) = 200 x^{199} + \cos x$
For $x \in (0, 1)$ and $(0, \frac{π}{2})$

$\cos x$ and $x$ are both positive $\Rightarrow$ $f (x)$ is increasing

For $x \in (\frac{π}{2}, π), x > 1$ hence $200 x^{199} > \cos x$ $\Rightarrow$ $f (x)$ is increasing

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