Monotonically Increasing Functions

Q.

How Do You Check If A Function Is Monotonically Increasing?

Q.

Prove the following:

${\text{cos}}^2$ 2x - ${\text{cos}}^2$ 6x = sin 4x sin 8x

Q.

Let 4 $x^2$ - 4(a - 2) x + a - 2 = 0, a R be a quadratic equation with real roots. Atleast one root of this equation lies in (0, 0.5) if

1. (2, 3)

2. (, 2)(3, )

3. (, 4)(5, )

4. (4, 5)

Q.

In Triangle ABC the value of $\frac{r_1+r_2}{1+cosC}$ is always equal to

1. 2r

2. 

3. 2R

4. 

Q. If $f\left(x\right)=30-2x-x^3$, then the number of positive integral value(s) of $x$ satisfying $f\left(f\right(f\left(x\right)\left)\right)>f\left(f\right(-x\left)\right)$ is
(correct answer + 1, wrong answer - 0.25)

1. $3$
2. $4$
3. $1$
4. $2$

Q. If $f\left(x\right)=x{e}^{x(1-x)}$, then f(x) is

1. increasing on $[\frac{-1}{2},1]$
2. decreasing on R
3. increasing on R
4. decreasing on $[\frac{-1}{2},1]$

Q. Let $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$. Then $f$ is an increasing function in the interval :

1. $[\frac{5\pi }{8},\frac{3\pi }{4}]$
2. $[\frac{\pi }{2},\frac{5\pi }{8}]$
3. $[0,\frac{\pi }{4}]$
4. $[\frac{\pi }{4},\frac{\pi }{2}]$