Test for Monotonicity about a Point

Q. If $2\le x\le 4,$ then the maximum value of
$f\left(x\right)=(x-2{)}^6(4-x{)}^5$ is

1. ${\left(\frac{11}{12}\right)}^6{\left(\frac{11}{10}\right)}^5$
2. ${\left(\frac{2}{11}\right)}^6{\left(\frac{10}{9}\right)}^5$
3. ${\left(\frac{12}{11}\right)}^6{\left(\frac{10}{11}\right)}^5$
4. ${\left(\frac{2}{11}\right)}^6{\left(\frac{10}{11}\right)}^5$

Q. $f\left(x\right)$ is cubic polynomial with $f\left(2\right)=18$ and $f\left(1\right)=-1.$ Also $f\left(x\right)$ has local maxima at $x=-1$ and $f\text{'}\left(x\right)$ has local minima at $x=0$, then

1. the distance between $(-1,2)$ and $(a,f(a\left)\right)$, where $x=a$ is the point of local minima is $2\sqrt{5}$
2. $f\left(x\right)$ is increasing for $x\in [1,2\sqrt{5}]$
3. $f\left(x\right)$ has local minima at $x=1$
4. the value of $f\left(0\right)=15$

Q. Prove that the function given by $f\left(x\right)=\cos x$ is
$\left(a\right)$ decreasing in $(0,\pi )$
$\left(b\right)$ increasing in $(\pi ,2\pi )$, and
$\left(c\right)$ neither increasing nor decreasing in $(0,2\pi )$

Q.

Evaluate, when

(i) n = 6, r = 2 (ii) n = 9, r = 5

Q. Let $y=min\{x,x^2,x^3\}$. At how many points in the interval $(-1,1]$, $y$ is not differentiable?

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

Q.

The function f (x) = -3x + 12 on R.is

1. decreasing

2. increasing

3. neither increasing or decreasing

4. strictly decreasing

Q. f(x) = ln(x) is neither increasing nor decreasing at x = 1.

Q. If $f\left(x\right)$ be a polynomial function satisfying
$f\left(x\right)f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$ and $f\left(4\right)=65$. Then, find $f\left(6\right)$.

Q.

f(x) = sinx is decreasing about $x=\pi$.

1. False

2. True

Q.

The function f (x) = a^x, 0 < a < 1 is

1. increasing

2. strictly decreasing on R

3. neither increasing or decreasing

4. decreasing

Q.

Find the point(s) where the function f(x) = $x^3$ - 3x + 2 is increasing

1. x = 0

2. None of these

3. x = 2

4. x = 1

Q.

Findf(x), where f(x) =

Q. Find the intervals in which the function f given by f ( x ) = 2 x 3 − 3 x 2 − 36 x + 7 is (a) strictly increasing (b) strictly decreasing

Q. The function $f\left(x\right)=\frac{x}{1+\left|x\right|}\mathrm{is}$
(a) strictly increasing
(b) strictly decreasing
(c) neither increasing nor decreasing
(d) none of these

Q. Let $f\left(x\right)$ be a polynomial of degree 6, which satisfies $\underset{x\to 0}{lim}{(1+\frac{f\left(x\right)}{x^3})}^{1/x}=e^2$ and local maximum at $x=1$ and local minimum at $x=0$ and $2$, then $5f\left(3\right)$ is equal to

Q.

Show that the function given by f(x) = 3x + 17 is strictly increasing on R.

Q.

1. f(x) is increasing on (e, ∞) and decreasing on (0, e)

2. f(x) is increasing on (0, ∞) and decreasing on (-e, 0) - {-1}

3. f(x) is increasing on ( -e, ∞) and decreasing on ( - ∞, - e) - {1}

4. f(x) is increasing on (e, ∞) and decreasing on (0, e) - {1}

Q. The function
(a) strictly increasing
(b) strictly decreasing
(c) neither increasing nor decreasing
(d) none of these

Q. If $f\left(x\right)=x^2+2bx+2c^2$ and $g\left(x\right)=-x^2-2cx+b^2$ are such that $f(x{)}_{min}>g(x{)}_{max},$ then the relation between $b$ and $c$ is $($where $b,c\in R)$

1. Not possible
2. $0<c<\frac{b}{2}$
3. $|c|<|b|\sqrt{2}$
4. $|c|>|b|\sqrt{2}$

Q.

Prove that the function given by is increasing in R.

Q. If $f\left(x\right)=\{\begin{array}{cc}2x+3,& \text{if}x<0\\ x^2-1,& \text{if}x\ge 0\end{array}$

and $g\left(x\right)=\{\begin{array}{cc}{x}^3,& \text{if}x<1\\ 3x-5,& \text{if}x\ge 1\end{array}$,

then the function $f\left(g\right(x\left)\right)$ is

1. one-one and onto
2. many one and onto
3. one-one and into
4. many one and into

Q. Solve this:

$44.\mathrm{The}\mathrm{function}\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{x}}{{\mathrm{e}}^{\mathrm{x}}-1}+\frac{\mathrm{x}}{2}+1,\mathrm{is}$
(1) An even function (2) An odd function
(3) A periodic function (4) Neither an even nor odd function

Q. If $\frac{d}{dx}$ (F(x) ) = f(x), then which of the following is/are antiderivative of f(x) ?

1. F(x)
2. kF(x), where k is a constant
3. F(x) + k
4. F(x) - k

Q. $f\left(x\right)$ is cubic polynomial with $f\left(2\right)=18$ and $f\left(1\right)=-1.$ Also $f\left(x\right)$ has local maxima at $x=-1$ and $f\text{'}\left(x\right)$ has local minima at $x=0$, then

1. the distance between $(-1,2)$ and $(a,f(a\left)\right)$, where $x=a$ is the point of local minima is $2\sqrt{5}$
2. $f\left(x\right)$ is increasing for $x\in [1,2\sqrt{5}]$
3. $f\left(x\right)$ has local minima at $x=1$
4. the value of $f\left(0\right)=15$

Q.

Find the points where the function f(x) = $x^3$ - 3x + 2 is increasing

1. x = 1

2. x = 2

3. None of these

4. x = 0

Q. Let $f\left(x\right)=\frac{a{x}^2+2(a+1)x+9a+4}{x^2-8x+32}$ and $g\left(x\right)=x-x^2-1.$ If $f\left(x\right)$ is less than $M+\frac{3}{4}$ for all $x\in R,$ where $M$ is the maximum value of $g\left(x\right),$ then set of values of $a$ lies in

1. $(-\infty ,\frac{-1}{2}]$
2. $(-\infty ,\frac{-1}{2})$
3. $(-\infty ,-5)$
4. $(-\infty ,0)$

Q. If $f:R\to R$ and $g:R\to R$ are two functions such that -
$f\left(x\right)+f^{\prime \prime }\left(x\right)=-xg\left(x\right)f^{\prime }\left(x\right)$ , $g\left(x\right)>0$ and $f\left(x\right)\in R,f^{\prime }\left(0\right)\ne 0$ . Then the function $f^2\left(x\right)+\left(f^{\prime }\right(x){)}^2$ has

1. minima at $x=0$
2. maxima at $x=0$
3. point of inflexion at $x=0$
4. none of the above

Q. Let $f\left(x\right)=\{\begin{array}{cc}x+2,& -1\le x<0\\ 1,& x=0\\ \frac{x}{2},& 0<x\le 1\end{array}$
Then in $[-1,1],$ this function has

1. a minimum
2. a maximum
3. either a maximum or a minimum
4. neither a maximum nor a minimum

Q. Let $x_1,x_2,.....x_n$ be values taken by a variable $X$ and $y_1,y_2,.....y_n$ be the values taken by variable $Y$ such that $y_i={ax}_i+b;i=1,2,....n$. Then,

1. $Var\left(Y\right)=a^{2}Var\left(X\right)$
2. $Var\left(X\right)=Var\left(X\right)+b$
3. none of these
4. $Var\left(X\right)=a^{2}Var\left(Y\right)$

Q.

Find the point(s) where the function f(x) = $x^3$ - 3x + 2 is increasing

1. x = 2

2. None of these

3. x = 0

4. x = 1