List of top Mathematics Questions on Increasing and Decreasing Functions asked in JEE Main

Given: \(5f (x) 4f(\frac 1x) = x^2 - 4 \) & \(y = 9f(x) × x^2\) If y is strictly increasing, then find interval of \(x\).
Let \(f:→R→(0,∞)\) be increasing function such that \(Lt_{x→∞}\frac {f(7x)}{f(x)}=1\), then \(Lt_{x→∞}[\frac {f(5x)}{f(x)}-1]\) is equal to
In which interval the function \(f(x) = \frac {x}{(x^2-6x-16)}\) is increasing?

Let $\alpha \in(0,1)$ and $\beta=\log _e(1-\alpha)$ Let $P_n(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\ldots+\frac{x^n}{n}, x \in(0,1)$ Then the integral $\int\limits_0^\alpha \frac{t^{50}}{1-t} d t$ is equal to

Let \(g : (0, ∞) →R\) be a differentiable function such that \(∫(\frac {x(cos⁡x−sin⁡x)}{e^x+1} + \frac {g(x)(e^x+1−xe^x)}{(e^x+1)2})dx = \frac {xg(x)}{e^x+1}+c,\) for all \(x>0\), where c is an arbitrary constant. Then:
The equation $x log x = 2 - x$ is satisfied by at least one value of $x$ lying between $1$ and $2$. The function $f(x) = x log x$ is an increasing function in $[1,2]$ and $g (x)=2-x$ is a decreasing function in $[1, 2]$ and the graphs represented by these functions intersect at a point in $[1,2]$