First Derivative Test for Local Maximum

Q.

The domain of the function $f\left(x\right)=\frac{\sqrt{-{\log }_{0.3}\left(x-1\right)}}{\sqrt{-x^2+2x+8}}$is

1. $\left(1,4\right)$

2. $\left(-2,4\right)$

3. $[2,4)$

4. None of these

Q. Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:

1. $12.5$
2. $10$
3. $25$
4. $30$

Q.

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

1. $\left[-1,-\frac{1}{\sqrt{2}}\right]\cup \left[\frac{1}{\sqrt{2}},1\right]$

2. $\left[-1,1\right]$

3. $\left[-\infty ,-\frac{1}{2}\right]\cup \left[\frac{1}{\sqrt{2}},\infty \right]$

4. $\left[\frac{1}{\sqrt{2}},1\right]$

Q. Let $g_i:[\frac{\pi }{8},\frac{3\pi }{8}]\to R,i=1,2$ and $f:[\frac{\pi }{8},\frac{3\pi }{8}]\to R$ be functions such that $g_1\left(x\right)=1,g_2\left(x\right)=|4x-\pi |$ and $f\left(x\right)={\sin }^{2}x,$ for all $x\in [\frac{\pi }{8},\frac{3\pi }{8}].$
Define $S_i={\int }_{\frac{\pi }{8}}^{\frac{3\pi }{8}}f\left(x\right)\cdot g_i\left(x\right)dx,i=1,2$

The value of $\frac{48S_2}{{\pi }^2}$ is

Q.

The value of $a^{{\log }_b\left(x\right)}$ where $a=0.2$, $b=\sqrt{5}$, $x=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+....\infty$ is

1. $1$

2. $2$

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

4. $4$

Q. A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $V{\text{mm}}^3,$ has a $2$ mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness $2$ mm and is of radius equal to the outer radius of the container.
If the volume of the material used to make the container is minimum when the inner radius of the container is $10$ mm, then the value of $\frac{V}{250\pi }$ is

Q. y=A sin(wt-kx) Find dy/dx.

Q.

Let f(x) =[n+p sin x], x belongs to (0, π), n belongs to Z , p is a prime number and [x] is greatest integer less than or equal to x .The number of points at which f(x) is not differentiable is

A. p

B. p-1

C. 2p+1

D. 2p-1

Q. Let $p\left(x\right)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. if $p\left(1\right)=6,p\left(3\right)=2$, then $p^{\prime }\left(0\right)$ is

Q. The period of sin πx/2+ cos πx/3 is

Q. Find the number of distinct real tangent that can be drawn from (0, -2) to parabola y^2=4x . Also find slope of tangents

Q. Find the degree and order of the differential equation y''+sin(y"')=0

Q.

Find the amplitude of the complex number sin(6π/5) + i (1- cos(6π/5))

Q. Let $f:[0,\infty )\to [0,3]$ be a function defined by
$f\left(x\right)=\{\begin{array}{cc}max\{\sin t:0\le t\le x\},& 0\le x\le \pi \\ 2+\cos x,& x>\pi \end{array}$
Then which of the following is true?

1. $f$ is continuous everywhere but not differentiable exactly at two points in $(0,\infty )$
2. $f$ is continuous everywhere but not differentiable
   exactly at one point in $(0,\infty )$
3. $f$ is differentiable everywhere in $(0,\infty )$
4. $f$ is not continuous exactly at two points in $(0,\infty )$

Q. Consider the function $f\left(x\right)=\text{sgn}(x-1)$ and $g\left(x\right)={\cot }^{-1}[x-1],$ where $[.]$ denotes the greatest integer function.

Statement $1:$ The function $F\left(x\right)=f\left(x\right)\cdot g\left(x\right)$ is discontinuous at $x=1.$
Statement $2:$ If $f\left(x\right)$ is discontinuous at $x=a$ and $g\left(x\right)$ is also discontinuous at $x=a,$ then the product function $f\left(x\right)\cdot g\left(x\right)$ is discontinuous at $x=a.$

1. Both the statements are true and Statement $2$ is the correct explanation of Statement $1.$
2. Both the statements are true and Statement $2$ is not the correct explanation of Statement $1.$
3. Statement $1$ is true and Statement $2$ is false.
4. Statement $1$ is false and Statement $2$ is true.

Q. Let the equation of a curve be x=a(theta+sintheta), y=a(1-costheta). If theta changes at a constant rate k then the rate of change of the slope of the tangent to the curve at theta=/3 is

Q. Difference between the greatest and the least values of the function $f\left(x\right)=x(\ln x-2)$ on $[1,e^2]$ is

1. $e$
2. $e^2$
3. $2$
4. $1$

Q. A rectangular surface has length 4661 metres and breath 3318 metres. On this area square tiles are be put. Find the maximum length of such tiles

Q. Let $f:R\to R$ be given by
$f\left(x\right)=\{\begin{array}{c}{x}^5+5x^4+10x^3+10x^2+3x+1,x<0;\\ x^2-x+1,0\le x<1;\\ \frac{2}{3}{x}^3-4x^2+7x-\frac{8}{3},1\le x<3;\\ (x-2){\log }_e(x-2)-x+\frac{10}{3},x\ge 3.\end{array}$
Then which of the following option is/are correct?

1. $f$ is increasing on $(-\infty ,0)$
2. $f^{\prime }$ has a local maximum at $x=1$
3. $f$ is onto
4. $f^{\prime }$ is NOT differentiable at $x=1$

Q.

Period of the function 2 sin⁴x + 3 cos⁴x is

Q.

The ratio of the length of a school ground to its width is 5: 2. Find the length if the width of the ground is 50 m.

Q. How to find the maxima and minima of any function …with some example.

Q.

Find the slope of the tangent to the curve y = 3x⁴ − 4x at x = 4.

Q. If $x=1$ is a critical point of the function $f\left(x\right)=(3x^2+ax-2-a)e^x$, then

1. $x=1$ is a local minima and $x=-\frac{2}{3}$ is a local maxima of $f.$
2. $x=1$ is a local maxima and $x=-\frac{2}{3}$ is a local minima of $f.$
3. $x=1$ and $x=-\frac{2}{3}$ are local minima of $f.$
4. $x=1$ and $x=-\frac{2}{3}$ are local maxima of $f.$

Q. Statement-1: $e^{\pi }>{\pi }^e$
Statement-2: ${\pi }^{e\pi }>e^{e\pi }$

1. Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1
2. Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation of Statement-1
3. Statement-1 is false, Statement-2 is true
4. Statement-1 is true, Statement-2 is false

Q. Integrate(sin 2x+cos 4x)dx with upper and lower limits as /4 and 0 respectively

Q.

Consider a right triangle with a hypotenuse of fixed length $45\text{cm}$ and variable legs of lengths $x$ and $y$ respectively.

If $x$ increases at the rate of $2\text{cm/min}$, how fast is $y$ changing when $x$ is $4\text{cm}$?

Q.

Fine common tangent to hyperbola 9x^2 - 9y^2 = 8 and the parabola y^2 = 32x

9x+3y= 8

9x-3y= -8

9x-3y= -4

9x-3y= 8

Q.

If 32 sin^6(x)=10-15 cos(2x) + b cos(4x) + a cos(6x), then how do we prove that tan(x)tan(5x)=(a+b-5)/(a-b+7)?

Q.

A differentiable function f(x) will have a local maximum at x = c if -

1. f’(c) = 0 , f’(c-h) < 0 & f’(c+h) > 0

2. f’(c) = 0 , f’(c-h) > 0 & f’(c+h) < 0

3. f’(c) = 0

4. None of the above