Monotonicity

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

The total number of distinct $x\in[0,1]$ for which $\displaystyle\int_0^x\frac{t^2}{1+t^4}\,dt=2x-1$ is

Let $f(x)=\displaystyle\lim_{n\to\infty}\left(\frac{n^n(x+n)\left(x+\frac n2\right)\cdots\left(x+\frac nn\right)}{n!\,(x^2+n^2)\left(x^2+\frac{n^2}{4}\right)\cdots\left(x^2+\frac{n^2}{n^2}\right)}\right)^{x/n}$, for all $x>0$. Then

Let $f(x)=\sin^4x+\cos^4x$. Then f is an increasing function in the interval :

Let $y(x)$ be a solution of the differential equation $(1+e^x)y'+ye^x=1$. If $y(0)=2$, then which of the following statements is (are) true?

The correct statement(s) is(are)

Let k and K be the minimum and the maximum values of the function $f(x)=\dfrac{(1+x)^{0.6}}{1+x^{0.6}}$ in [0, 1] respectively, then the ordered pair (k, K) is equal to :

Let $f:(0,\infty)\to\mathbb R$ be given by
$$f(x)=\int_{1/x}^{x}e^{-\left(t+\frac1t\right)}\frac{dt}t.$$
Then

Let $f:\left(-\frac\pi2,\frac\pi2\right)\to\mathbb R$ be given by
$$f(x)=\left(\log(\sec x+\tan x)\right)^3.$$
Then

If $f(x)=\left(\dfrac35\right)^x+\left(\dfrac45\right)^x-1$, $x\in\mathbf R$, then the equation $f(x)=0$ has :

Let $f$ and $g$ be two differentiable functions on $\mathbb R$ such that $f'(x)>0$ and $g'(x)<0$, for all $x\in\mathbb R$. Then for all $x$ :

Let $f:\left[\frac12,1\right]\to\mathbb R$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f'(x)<2f(x)$ and $f\left(\frac12\right)=1$. Then the value of $\displaystyle\int_{1/2}^1f(x)\,dx$ lies in the interval

The number of points in $(-\infty,\infty)$, for which $x^2-x\sin x-\cos x=0$, is

Which of the following is true for $0<x<1$?

If the function $e^{-x}f(x)$ assumes its minimum in the interval $[0,1]$ at $x=\frac14$, which of the following is true?

The function $f:[0,3]\to[1,29]$, defined by $f(x)=2x^3-15x^2+36x+1$, is

Which of the following is true?

The number of distinct real roots of $x^4-4x^3+12x^2+x-1=0$ is

Match the statements given in Column I with the intervals/union of intervals given in Column II

The real number $s$ lies in the interval