Application of Derivatives

The minimum distance of a point on the curve$y=x^{2}-4$ from the origin is

Let $f\left(x\right)=3^{\left(x^{2}-2\right)^{3}+4},x\in\,R$. Then which of the following statements are true? $P:x=0$is a point of local minima of$f$ $Q:x=\sqrt{2}$is a point of inflection of$f$ $R:f^{'}$is increasing…

Let $a$be a real number such that the function$f(x)=ax^{2}+6x-15,x\in\,R$ is increasing in$\left(-\infty\,,\frac{3}{4}\right)$ and decreasing in $\left(\frac{3}{4},\infty\,\right)$. Then the function…

Let $f:\left(a,b\right)\rightarrow\,R$ be twice differentiable function such that $f\left(x\right)=\int\,_{a}^{x}g\left(t\right)dt$ for a differentiable function $g\left(x\right).$ If $f\left(x\right)=0$has exactly five…

If $S_{1}$ and $S_{2}$ are respectively the sets of local minimum and local maximum points of the function, $f\left(x\right)=9x^{4}+12x^{3}-36x^{2}+25,x\in\,R,$ then

The function $f\left(x\right)=x^{3}-6x^{2}+ax+b$ is such that $f\left(2\right)=f\left(4\right)=0$. Consider two statements: $\left(S_{1}\right)$ there exists $x_{1},x_{2}\in\,\left(2,4\right),x_{1}$$\left(S_{2}\right)$$…

If$m$ and $n$respectively are the number of local maximum and local minimum points of the function $f\left(x\right)=\int\,_{0}^{x^{2}}\frac{t^{2}-5t+4}{2+e^{t}}dt$, then the ordered pair $\left(m,n\right)$ is equal to

The function $f\left(x\right)=2x+3x^{\frac{2}{3}},x\in\,R$, has

Let $f\left(x\right)=3\sin\,^{4}x+10\sin\,^{3}x+6\sin\,^{2}x-3$,$x\in\,\left[-\frac{\pi\,}{6},\frac{\pi\,}{2}\right]$.Then, $f$ is :

Let $f:R\rightarrow\,R$ be defined as$f\left(x\right)=\left{\begin{matrix} -\frac{4}{3}x^{3}+2x^{2}+3x, & x>0 \\ 3xe^{x}, & x\leq\,0 \end{matrix}.\right$Then$f$isincreasing function in the interval

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

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $tan^{-1}\left(\frac{1}{2}\right).$ Water is poured into it at a constant rate of $5cubicm/min.$Then the rate $($in $m/min),$at…