Condition for Strictly Decreasing

Q. Consider the function $f\left(x\right)={\sin }^{5}x+{\cos }^{5}x-1,$ $x\in [0,\frac{\pi }{2}].$ Which of the following is (are) CORRECT?

1. $f$ is strictly decreasing in $[0,\frac{\pi }{4}]$
2. $f$ is strictly increasing in $[\frac{\pi }{4},\frac{\pi }{2}]$
3. There exists a number $c$ in $(0,\frac{\pi }{2})$ such that $f^{\prime }\left(c\right)=0$
4. The equation $f\left(x\right)=0$ has only two roots in $[0,\frac{\pi }{2}]$

Q.

The domain of the function $f\left(x\right)=\sqrt{{\cos }^{-1}\left(\frac{1-\left|x\right|}{2}\right)}$ is

1. $(-3,3)$

2. $[-3,3]$

3. $(-\infty ,-3)\cup (3,\infty )$

4. $(-\infty ,-3]\cup [3,\infty )$

Q.

Let $f:R\to R$ be defined as

$$\begin{gathered} f\left(x\right)=\left\{x+a,x<0 \\ \left|x-1\right|,x\le 0\right\} \\ g\left(x\right)=\left\{x+1,x<0 \\ {\left(x-1\right)}^2+b,x\ge 0\right\} \end{gathered}$$

where $a,b$ are non-negative real numbers. If $\left(gof\right)\left(x\right)$ is continuous for all $x\in R,$ then $a+b$ is equal to:

Q.

Which of the following is the graph of sgn (x), where sgn (x) represents the signum function?

1. 

2. 

3. 

4. 

Q. All the functions with straight line graphs are either strictly increasing functions or strictly decreasing functions.

1. F

Q.

A strictly increasing function defined on [ a, b ] will have its global maximum at x =b..

1. True

2. Flase

Q.

A strictly increasing function defined on [ a, b ] will have its global minimum at x =a.

1. True

2. Flase

Q. Let $f$ be a twice differentiable function in $(-\infty ,\infty )$ such that $f^{\prime \prime }\left(x\right)<0\forall x\in R.$ If $g\left(x\right)=f\left(x\right)+f(1-x)$ and $g^{\prime }\left(\frac{1}{4}\right)=0,$ then

1. $g\left(x\right)$ is increasing in $(-\infty ,0)$
2. $g\left(x\right)$ attains local minimum at $x=\frac{1}{2}$
3. $g^{\prime \prime }\left(x\right)=0$ has at least two roots in $[0,1]$
4. $g^{\prime }\left(x\right)<0$ for all $x\in (\frac{1}{2},\infty )$

Q. Objective function of a LPP is
(a) a constraint
(b) a function to be optimized
(c) a relation between the variables
(d) none of these

Q. The antiderivative of the function $(3x+4)|\sin x|$, when $0<x<\pi$, is given by

1. $3\sin x+(3x+4)\cos x+c$
2. $-3\sin x+(3x+4)\cos x+c$
3. $3\sin x-(3x+4)\cos x+c$
4. $3\cos x+(3x+4)\sin x+c$

Q. The value of 2$\times 2{\sin }^2\frac{\pi }{6}+2{\text{cosec}}^2\frac{7\pi }{6}{\cos }^2\frac{\pi }{3}$

Q.

On the interval $[0,1],$ the function $x^{25}{(1-x)}^{75}$ takes its maximum value at the point

1. $0$

2. $\frac{1}{2}$

3. $\frac{1}{3}$

4. $\frac{1}{4}$

Q. Solve:

Q. All the functions with straight line graphs are either strictly increasing functions or strictly decreasing functions.

1. F

Q. The function $f\left(x\right)=-1+\frac{x^2}{2}+\cos x$ has , where $x\in [-\frac{\pi }{2},\frac{\pi }{2}]$

1. The minimum value $-2$
2. No extremum at $x=0$
3. The minimum value $0$
4. None of these

Q. $2x+3y=sinx$ find $\frac{dy}{dx}$

Q. The integral ${\int }_0^{\pi }xf\left(\sin x\right)dx$ is equal to

1. $\frac{\pi }{2}{\int }_0^{\pi }f\left(\sin x\right)dx$
2. $\pi {\int }_0^{\pi /2}f\left(\cos x\right)dx$
3. $\frac{\pi }{4}{\int }_0^{\pi }f\left(\sin x\right)dx$
4. $\pi {\int }_0^{\pi /2}f\left(\sin x\right)dx$

Q. Consider the function $f\left(x\right)={\sin }^{5}x+{\cos }^{5}x-1,$ $x\in [0,\frac{\pi }{2}].$ Which of the following is (are) CORRECT?

1. $f$ is strictly decreasing in $[0,\frac{\pi }{4}]$
2. $f$ is strictly increasing in $[\frac{\pi }{4},\frac{\pi }{2}]$
3. There exists a number $c$ in $(0,\frac{\pi }{2})$ such that $f^{\prime }\left(c\right)=0$
4. The equation $f\left(x\right)=0$ has only two roots in $[0,\frac{\pi }{2}]$

Q. The value of ${\int }_0^{\pi }|\cos x{|}^{3}dx$.

1. $\frac{4}{3}$
2. $0$
3. $\frac{-4}{3}$
4. $\frac{2}{3}$

Q. How do you prove $\left(\frac{cosx}{1+sinx}\right)+\left(\frac{cosx}{1-sinx}\right)=2secx$ ?

Q. If $\int f\left(x\right)\sin x\cos xdx=\frac{1}{2(b^2-a^2)}\ln f\left(x\right)+c$, then $\frac{1}{f\left(x\right)}$ is equal to

Q.

The above graph of a function is:

1. Decreasing

2. Strictly Increasing

3. Neither Increasing nor Decreasing

4. Increasing

Q. lf $y_k$ is the $k^{th}$ derivative of $y$ w.r.t. $x$ and $y=\cos \left(\sin x\right)$ then $y^{\prime }\sin x+y^{\prime \prime }\cos x=$

1. $-y{\sin }^{3}x$
2. $-y{\cos }^{3}x$
3. $-y{\cos }^{2}x$
4. $y{\sin }^{3}x$

Q. Let $\alpha =\underset{x\in R}{max}\{8^{2\sin 3x}\cdot 4^{4\cos 3x}\}$ and $\beta =\underset{x\in R}{min}\{8^{2\sin 3x}\cdot 4^{4\cos 3x}\}.$
If $8x^2+bx+c=0$ is a quadratic equation whose roots are ${\alpha }^{1/5}$ and ${\beta }^{1/5}$, then the value of $c-b$ is equal to

1. $43$
2. $42$
3. $50$
4. $47$

Q. If $K={\left(\sin y\right)}^{\sin \left(\frac{\pi }{2}x\right)}$, then at $x=1,\frac{dy}{dx}=$

1. $p$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{2}\log K$
4. $0$

Q. ${\int }_0^{2\pi }{e}^x.sin(\frac{\pi }{4}+\frac{x}{2})dx$

Q. The inequality.
$maxf\left(x\right)<0.77[-\pi ,\pi ]$ holds for the function $f\left(x\right)=\left(sinx\right)\left(sin2x\right).$

1. True
2. False

Q.

What is the value of $a+b$ when function $f\left(x\right)$ given as

$f\left(x\right)=\left\{\begin{array}{l}\sin x-e^{x}x<0\\ a+\left|-x\right|0\le x<1\\ 2x-bx\ge 1\end{array}\right.$

is continuous in $(-\infty ,1]$.

Q. The function $f\left(x\right)=-1+\frac{x^2}{2}+\cos x$ has , where $x\in [-\frac{\pi }{2},\frac{\pi }{2}]$

1. None of these
2. No extremum at $x=0$
3. The minimum value $-2$
4. The minimum value $0$

Q. If $f\left(x\right)=\frac{{(a+x)}^2\sin (a+x)-a^2\sin a}{x},x\ne 0$, the value of $f\left(0\right)$ so that f is continuous at $x=0$ is

1. $a^2\cos a+a\sin a$
2. $a^2\cos a+2a\sin a$
3. $2a^2\cos a+a\sin a$
4. none of these