Monotonicity in an Interval

Q.

The function$f:X\to Y$defined by $f\left(x\right)=\sin x$ is one-one but not onto if$X$and$Y$ are respectively equal to.

1. $\mathrm{ℝ}$ and $\mathrm{ℝ}$

2. $[0,\mathrm{\pi }]$and$[0,1]$

3. $\left[0,\frac{\mathrm{\pi }}{2}\right]$and$\left[-1,1\right]$

4. $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$and $[-1,1]$

Q. Evaluate tan inverse[(3sin2x)/(5+3cos2x)]+tan inverse[(1/4)tanx], where -pi/2

Q. The function $f\left(x\right)=\frac{2x^2-1}{x^4},x>0$, decreases in the interval

1. $[1,\infty )$
2. $\left(1,\infty \right)$
3. $\left(0,\infty \right)$
4. $[0,\infty )$

Q.

All the numbers that can be formed using 1, 2 , 3, 4, 5 are arranged in the decreasing order then the rank of 35 241 is

Q. If t be a real number satisfying the equation $2t^3-9t^2+30-a=0$, then the value of the parameter $a$ for which the equation $x+\frac{1}{x}=t$, gives six real and distinct values of $x$.

1. $-11$
2. $10$
3. $21$
4. No value exists

Q. Which of the following is/are true, For $f\left(x\right)=\frac{ln\left(lnx\right)}{lnx}$

1. (ln28)^ln30 < (ln30)^ln28
2. (ln 2.1)^ln2.2 > (ln 2.2)^ln2.1
3. (ln 4)^ln5 < (ln5)^ln4
4. (ln30)^ln31 < (ln31)^ln30

Q. The function $f\left(x\right)=4{\sin }^{3}x–6{\sin }^{2}x+12\mathrm{sinx}+100$ is strictly

1. decreasing in $(\frac{\pi }{2},\frac{3\pi }{2})$
2. decreasing in $(\frac{\pi }{2},\pi )$
3. increasing in $(\frac{-\pi }{2},\frac{\pi }{2})$
4. increasing in $(\frac{\pi }{2},\frac{3\pi }{2})$

Q.

Find the intervals in which the following functions are strictly increasing or decreasing:

(a) x² + 2x − 5 (b) 10 − 6x − 2x²

(c) −2x³ − 9x² − 12x + 1 (d) 6 − 9x − x²

(e) (x + 1)³ (x − 3)³

Q.

The maximum and the minimum value of 3x⁴ – 8x³ + 12x² – 48x + 1 on the interval [1, 4]

1. -49, 258

2. -58, 257

3. -40, 257

4. -48, 258

Q. The interval of increase of the function f(x)=x - e^{​​​​​​x} + tan(2π/7) is

Q. $\int \sqrt{9-x^2}dx$

Q.

1. 1

2. zero

3. w

4. -1

Q. Show that the function given by $f\left(x\right)=7x-3$ is strictly increasing on $R$.

Q.

Interval in which {x} is monotonically increasing function, where { } is fractional part function -

1. [0, 1)

2. [1 /2, 3 /2 ]

3. [1, 2)

4. [3 /2 , 5 /2]

Q.

What is mean by (-infinity, infinity)?

Q.

Which of the following statements are true about [x], greater integer Function?

1.$x-1)<[x]\le x$

2.$[-x]=1-\left[x\right]$, when x is NOT an integer

1. Only 1

2. Only 2

3. Both 1 and 2

4. None of these

Q. Prove that the function f given by f ( x ) = log cos x is strictly decreasing on and strictly increasing on

Q. Show that the function given by f ( x ) = sin x is (a) strictly increasing in (b) strictly decreasing in (c) neither increasing nor decreasing in (0, π)

Q.

If, then show that

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

Q. Let a function $f\left(x\right)=\frac{x^3}{3}-2x^2+3x$ on the set $A=\{x|x^2+8\le 6x\},$ then the minimum value of $f$ is

Q.

Functions which are not monotonic throughout their domains could be monotonic in an interval of their domain.

1. True

2. False

Q. Prove that f(x) = sinx + $\sqrt{3}$cosx has maximum value at x = $\frac{\mathrm{\pi }}{6}$. [NCERT EXEMPLAR]

Q. A positive function $f$ is such that $f(2-h)=4,f(2+h)=3$ as $h\to 0^{+}$ has a minimum at $x=2.$ Then the value of $f\left(2\right)$ can be

1. $2$
2. $3$
3. $5$
4. $4$

Q. Let $f\left(x\right)=a{x}^3+b{x}^2+cx+1$ have extrema at $x=\alpha ,\beta$ such that $\alpha \beta <0$ and $f\left(\alpha \right)f\left(\beta \right),0$. Then the equation $f\left(x\right)=0$ has.

1. Three equal real roots
2. One positive root if $f\left(\alpha \right)<0$ and $f\left(\beta \right)>0$
3. One negative root if $f\left(\alpha \right)>0$ and $f\left(\beta \right)>0$
4. Three distinct real roots

Q. Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8).

Q. If x > 1, then $2{\tan }^{-1}x+{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)$is equal to
(a) $4{\tan }^{-1}x$
(b) 0
(c) $\frac{\mathrm{\pi }}{2}$
(d) $\mathrm{\pi }$

Q. If $f\text{'}\text{'}\left(x\right)>0,\forall x\in R,f\text{'}\left(3\right)=0$ and $g\left(x\right)=f({\tan }^{2}x-2\tan x+4),0<x<\frac{\pi }{2}$, then $g\left(x\right)$ is increasing in

1. $(0,\frac{\pi }{4})$
2. $(\frac{\pi }{6},\frac{\pi }{3})$
3. $(0,\frac{\pi }{3})$
4. $(\frac{\pi }{4},\frac{\pi }{2})$

Q.

Find the intervals in which the function f given by f(x) = 2x³ − 3x² − 36x + 7 is

(a) strictly increasing (b) strictly decreasing

Q. Show that 2 tan⁻¹ x + sin⁻¹ $\frac{2x}{1+x^2}$ is constant for x ≥ 1, find that constant.