Increasing & decreasing practice · Mixed difficulty

Let ${\lambda ^*}$ be the largest value of $\lambda$ for which the function ${f_\lambda }(x) = 4\lambda {x^3} - 36\lambda {x^2} + 36x + 48$ is increasing for all $x \in R.$ Then ${f_\lambda }^*(1) + {f_\lambda }^*( - 1)$ is equal to:

The length of the longest interval, in which the function $3\sin x - 4{\sin ^3}x$ is increasing, is

If $f\left( x \right) = 3{x^4} + 4{x^3} - 12{x^2} + 12$, then $f\left( x \right)$ is

Let $f(x)=x \cos ^{-1}\left(\sin (-|x|), x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\right.$ Then,

The function $f(x)=x^3-3 x$, is

If $f(x)=x^{3}-a x^{2}+48 x+7$ is increasing then a lies between

Let $f:(1,3) \rightarrow R$ be a function given by $f(x)=\frac{x[x]}{x^2+1}$, where $[\cdot]$ denotes the greatest integer function, then range of $f(x)$ is

Let $f(x)$ be a function defined by
$f(x)=\left(a b-a^2-2\right) x-\int\limits_{0}^x\left(\cos ^4 t+\sin ^2 t-2\right) d t$

If $f(x)$ is a decreasing function for all $x \in R$ and $a \in R$, where $a$ is independent of $x$, then

Let $f(x)=\cos x-1+\left(x^2 / 2\right)$. Then

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

The function $f(x)=x^x$ decreases on the interval

Let $f\,:(a,b)\, \to R$ be twice differentiable function such that $f(x)\, = \,\int_a^x {g(t)\,dt}$ for a differentiable function $g(x)$. If $f(x)\, = \,0$ has exactly five distinct roots in $(a,b)$, then $g(x)\,g'(x)\, = \,0$ has at least

If $g(x)$ is a continuous function at $x=a$ such that $g(a) > 0$ and $f^{\prime}(x)=g(x)\left(x^2-a x+a^2\right)$ for all $x \in R$, then $f(x)$, is

Let $f(x)$ and $g(x)$ be increasing and decreasing functions respectively from $[0, \infty)$ to $[0, \infty)$, Let $h(x)=fog(x)$. If $h(0)=0$, then $h(x)$, is

$f(x)=\left(\frac{e^{2 x}-1}{e^{2 x}+1}\right)$ is

Consider the polynomial $f(x)=1+2 x+3 x^2+4 x^3.$ Let $s$ be the sum of all distinct real roots of $f(x)$. Then $s$ lies in the interval

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

The minimum value of the twice differentiable function $f(x)=\int\limits_0^x e^{x-t} f^{\prime}(t) \mathrm{dt}-\left(x^2-x+1\right) e^x, x \in \mathbb{R}$, is

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

Let $f(x)=x^{3}+6 x^{2}+a x+2$, if $(-3,-1)$ is the largest possible interval for which $f(x)$ is decreasing function, then $a=$

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

The maximum slope of the curve $y = \frac{1}{2}{x^4} - 5{x^3} + 18{x^2} - 19x$ occurs at the point

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

If $f(x)=x^3+b x^2+c x+d$ and $0 < b^2 < c$, then in $(-\infty, \infty)$

Let the function $f(x)=\tan ^{-1}(\sin x+\cos x)$ be defined on $[0,2 \pi]$. Then, $f(x)$ is

If $f\left( x \right) = k\;{x^3} - 9\;{x^2} + 9\;x + 3$ is increasing on $R$, then