Question 70
Consider the functions
$$
f(x) = \frac{\cos^2 x}{1 + \cos x + \cos^2 x} \text{ and } g(x) = k \tan x + (1-k) \sin x - x
$$
where $k$ is a real number.
a) Prove that $g'(x) = \frac{(1-\cos x)(k-f(x))}{f(x)}$.
b) Find the image of $f(x)$ if $x \in \left[0 ; \frac{\pi}{2}\right)$.
c) Find all $k$ such that $g(x) \geq 0$ for any $x \in \left[0 ; \frac{\pi}{2}\right)$.
$$
f(x) = \frac{\cos^2 x}{1 + \cos x + \cos^2 x} \text{ and } g(x) = k \tan x + (1-k) \sin x - x
$$
where $k$ is a real number.
a) Prove that $g'(x) = \frac{(1-\cos x)(k-f(x))}{f(x)}$.
b) Find the image of $f(x)$ if $x \in \left[0 ; \frac{\pi}{2}\right)$.
c) Find all $k$ such that $g(x) \geq 0$ for any $x \in \left[0 ; \frac{\pi}{2}\right)$.