Monotonically Decreasing Functions

Q. the value of (-1+root3i)^1008 +(-1-root3i)^1008 is

Q. harmonic mean of roots of the equation (5+\sqrt2)x^2-(4+\sqrt2)x+8+2\sqrt2=0 is: options 2, 4, 6, 8

Q. The set of all real values of $a$ for which the function $f\left(x\right)=(a+2)x^3-3a{x}^2+9ax-1$ decreases monotonically throughout for all real $x,$ is

1. $a<-2$
2. $a>-2$
3. $-3\le a<0$
4. $a\le -3$

Q. If $f\left(x\right)=x^3+4x^2+\mathrm{kx}+1$ is a monotonically decreasing function of x in the largest possible interval (-2, -2/3). Then .

1. k = 4
2. k = 2
3. k = -1
4. k has no real value

Q.

Constant functions are Monotonically increasing as well as monotonically decreasing functions

1. True

2. False

Q. THE RANGE OF VALUES OF a so that all the roots of the euation 2x^3-3x^2-12x+a=0 are real and distinct

Q. Let a and c be prime numbers and b an integer. Given that the quadratic equation ax^2+bx+c =0 has rational roots, show tha one of the root is independent of the coefficients. Find the 2 roots.

Q. Let $f:R\to R$ be a continuous decreasing function. A point $x_0\in R$ is said to be a fixed point of $f$ if $f\left(x_0\right)=x_0.$
The number of fixed points of $f\circ f\circ f$ equals

1. $0$
2. $1$
3. $2$
4. infinitely many

Q. The function $f\left(x\right)=\frac{ln(\pi +x)}{ln(e+x)}$ is

1. increasing on $(0,\infty )$
2. decreasing on $(0,\infty )$
3. increasing on $(0,\frac{\pi }{e})$, decreasing on $(\frac{\pi }{e},\infty )$
4. decreasing on $(0,\frac{\pi }{e})$, increasing on $(\frac{\pi }{e},\infty )$

Q. Find the domain of f(x)=root((3^x-2^x)/x).

Q. If $f\left(x\right)=64x^3+\frac{1}{x^3}$and α, β are the roots of $4x+\frac{1}{x}=3$. Then,
(a) f(α) = f(β) = −9
(b) f(α) = f(β) = 63
(c) f(α) ≠ f(β)
(d) none of these

Q. Let $f:R\to R$ be a continuous decreasing function. A point $x_0\in R$ is said to be a fixed point of $f$ if $f\left(x_0\right)=x_0.$
The number of distinct $3$-tuples $(x,y,z)$ satisfying the system
$x=f\left(y\right),y=f\left(z\right),z=f\left(x\right)$
equals

1. $0$
2. $1$
3. $2$
4. infinitely many

Q. Let $f:R\to R$ be a differentiable function for all values of x and has a property that f(x) and f '(x) have opposite signs for all values of x. Then,

1. f(x) is an increasing function.
2. f(x) is a decreasing function.
3. $f^2\left(x\right)$ is a decreasing function.
4. |f(x)| is an increasing function.

Q. Redefine the function f (x) = x − 2 + 2+ x , – 3 ≤ x ≤ 3

Q. Function $g\left(x\right)=9x-4\tan x,x\in (0,\frac{\pi }{2})$ will have

1. Local maximum at ${\cos }^{-1}(-\frac{2}{3})$
2. No local extremum
3. Local maximum at $x=\frac{1}{2}$
4. Local maximum at ${\cos }^{-1}\left(\frac{2}{3}\right)$

Q. lim(x tends to infinity) root(x+1)-root(x)