Application of Derivatives

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

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}$$\left(S_{2}\right)$$…

Let $(2 \alpha, \alpha)$ be the largest interval in which the function $f(t)=\frac{|t+1|}{t^{2}}, t2$, is $\_\_\_\_$

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that the quadratic equation $f(x) \mathrm{m}^{2}-2 f^{\prime}(x) \mathrm{m}+f^{\prime \prime}(x)=0$ in m, has two equal roots for every…

The distance from the origin, of the normal to the curve, $x=2\cos\,t+2t\sin\,t,y=2\sin\,t-2t\cos\,tatt=\frac{\pi\,}{4},$ is :

Let a curve $y=f\left(x\right),x\in\,\left(0,\infty\,\right)$pass through the points$P\left(1,\frac{3}{2}\right)$and$Q\left(a,\frac{1}{2}\right)$. If the tangent at any point $R(b,f(b))$to the given curve cuts the…

Let $P(x)$ be a real polynomial of degree $3$which vanishes at $x=-3$.Let $P(x)$ have local minimaat $x=1$, local maxima at $x=-1$and$\int\,_{-1}^{1}P\left(x\right)dx=18$, then the sum of all thecoefficients of the…

Let $f\left(x\right)$be a cubic polynomial with $f\left(1\right)=-10,f\left(-1\right)=6,$and has a local minima at $x=1,$and $f^{'}\left(x\right)$ has a local minima at $x=-1.$Then $f\left(3\right)$ is equal to .

Let$f\left(x\right)=\left|\begin{matrix} 1+\sin\,^{2}x & \cos\,^{2}x & \sin\,2x \\ \sin\,^{2}x & 1+\cos\,^{2}x & \sin\,2x \\ \sin\,^{2}x & \cos\,^{2}x & 1+\sin\,2x…

Let the normal at a point $P$ on the curve $y^{2}-3x^{2}+y+10=0$ intersect the $y$ -axis at $\left(0,\frac{3}{2}\right).$ If $m$ is the slope of the tangent at $P$ to the curve, then $\left|m\right|$ is equal to…

The minimum value of $\alpha\,$ for which the equation $\frac{4}{\sin\,x}+\frac{1}{1-\sin\,x}=\alpha\,$ has at least one solution in $\left(0,\frac{\pi\,}{2}\right)$ is______?