Test for Monotonicity in an Interval

Q. Let $f:R\to (0,1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0, 1)?

1. $x^9-f\left(x\right)$
2. $x-\underset{0}{\overset{\frac{\pi }{2}-x}{\int }}f\left(t\right)\cos tdt$
3. $e^x-\underset{0}{\overset{x}{\int }}f\left(t\right)\sin tdt$
4. $f\left(x\right)+\underset{0}{\overset{\frac{\pi }{2}}{\int }}f\left(t\right)\sin tdt$

Q.

Which of the following function are monotonic in the interval (0, 1) ?

1. $f\left(x\right)=x^2$

2. $f\left(x\right)=\frac{1}{x}$

3. $f\left(x\right)=x^3-x$

4. $f\left(x\right)=ln\left(x\right)$

Q. The function f(x)= $2x^3-9x^2+12x+4$ is decreasing in the interval(s)

1. $(-\infty ,1)$ and $(2,\infty )$
2. $(-\infty ,1)$ only
3. (1, 2)
4. $(2,\infty )$ only

Q.

$f\left(x\right)=1+\left[cosx\right]x,in0<x⩽\frac{\pi }{2}$

1. Has a minimum value 0
2. Has a maximum value 2

3. Is continuos in $[0,\frac{\pi }{2}]$

4. Is not differentiable at x = $\frac{\pi }{2}$