Increasing & decreasing practice · Medium

How many real solutions does the equation ${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$ have?

${\left( {1 + x} \right)^n} \le 1 + {x^n},$ where

Consider the following two statements.

I: If $f(x)$ and $g(x)$ are continuous and monotonic functions on the real line $\mathbb{R}$, then $f(x)+g(x)$ is also a monotonic function.

II: If $f(x)$ is a continuous decreasing function for all $x>0$ and $f(1)$ is positive, then $f(x)=0$ exactly at one value of $x$.
Then

The function $f(x)=2 \log (x-2)-x^2+4 x+1$

Set of all possible values of $k$ for which $f(x)=\sin x-\cos x-k x+b$ decreases for all real values of $x$, is:

In $[0,1], f$ is twice differentiable and $f^{\prime \prime}(x)<0$. Let

$\phi(x)=f(x)+f(1-x)$
for $x \in[0,1]$. Then

Let $f:[0,2] \rightarrow R$ be a twice differentiable function such that $f^{\prime \prime}(x)>0$, for all $x \in(0,2)$.If $\phi(x)=f(x)+f(2-x)$, then $\phi$ is

If $f\left( x \right) = \int \nolimits_{{x^2}}^{{x^2} + 1} {e^{ - {t^2}}}dt,$ then $f\left( x \right)$ increases on

The function $f\left( x \right) = \tan x - x$

If $f: R \rightarrow R$ be definend by $f(x)=2 x+\cos x$, then $f$

The function $f\left( x \right) = 2{x^3} - 3{x^2} + 90x + 174$ is increasing in the interval

If $f\left( x \right) = x{e^{x\left( {1 - x} \right)}}$, then $f\left( x \right)$ is

$f(x)=4 \log _e(x-1)-2 x^2+4 x+5, x>1$ which one of the following is NOT correct?

Let $f\left( x \right) = \int \limits_0^x \frac{{\cos t}}{t}dt$ Then, at $x = \left( {2n + 1} \right)\frac{\pi }{2},\;f\left( x \right)$ has

Let $f(x)$ and $g(x)$ be defined and differentiable for all $x \geq x_0$ and $f\left(x_{0})=g\left(x_0\right), f^{\prime}(x) > g^{\prime}(x)\right.$ for $x > x_0$, then

Let $g\left( x \right) = f\left( x \right) - 2{\left\{ {f\left( x \right)} \right\}^2} + 9{\left\{ {f\left( x \right)} \right\}^3}$ for all $x \in R.$ Then,

Let $f:(0,1) \rightarrow R$ be a function defined by $f(x)=\frac{1}{1-e^{-x}}$, and $g(x)=(f(-x)-f(x))$. Consider two statements

(I) $g$ is an increasing function in $(0,1)$

(II) $g$ is one-one in $(0,1)$
Then,

Let $f$ be differentiable on $\mathbb{R}$ and

$h(x)=f(x)-[f(x)]^2+[f(x)]^3$

Then

The function $f$ defined by $f(x)=x^3-3 x^2+5 x+7$, is :

The set of values of a for which the function
$f\left( x \right) = 2{e^x} - a{e^{ - x}} + \left( {2a + 1} \right)x - 3$
is increasing on $R$, is

The function $f(x)=2 x^2-\log |x|$ is

If $f(x)=\int\limits_{{x^2}}^{x^2+1} e^{-t^2} d t$, then $f(x)$ increases in

If $a, b$, $c$ be real, then $f(x)=\left|\begin{array}{ccc}x+a^2 & a b & a c \\ a b & x+b^2 & b c \\ a c & b c & x+c^2\end{array}\right|$, is decreasing on

The function $f(x)=\frac{4 x^3-3 x^2}{6}-2 \sin x+(2 x-1) \cos x$ is

If $f(x)=\left(a b+b^2+1\right) x+\int\limits_{0}^x\left(\cos ^4 \theta+\sin ^4 \theta\right) d \theta$ is an increasing function of $x$ for all $x \in R$ and $b \in R, b$ being independent of $x$, then

The function $f\left( x \right) = {x^{1/x}}$ is increasing in the interval

If $f(x)=x+\sin x, g(x)=e^{-x}, u=\sqrt{c+1}-\sqrt{c}$, $v=\sqrt{c}-\sqrt{c-1},(c > 1)$, then

Suppose $f(x)$ is a real-valued differentiable function such that $f(x) f^{\prime}(x)<0$ for all real $x$. Then

The number of integral values of a for which $f(x)=x^{3}+(a+2) x^{2}+3 a x+5$ is monotonic in $\forall x \in R$.

If $f(x)$ is a non-zero polynomial of degree four, having local extreme points at $x = - 1,\,0,\,1,$ then the set $S = \left\{ {x \in R:f(x) = f(0)} \right\}$ contains exactly