Application of Derivatives

Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $\left(1+x\left(\lambda^2-x^2\right)\right)$ satisfies $\frac{x^2+x+2}{x^2+5 x+6} < 0$, be $(\alpha, \beta)$. Then…

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…

Let $f:[-1,1]\rightarrow\,R$ be defined as $f(x)=ax^{2}+bx+c$for all $x\in\,[-1,1],$where $a,b,c\in\,R$ such that$f(-1)=2,f^{'}(-1)=1$and for $x\in\,(-1,1)$ the maximum value of $f^{"}(x)$ is $\frac{1}{2}.$…

A wire of length $36m$ is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the…

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

The number of points, where the curve$y=x^{5}-20x^{3}+50x+2$crosses the $x$-axis, is_____?

Let the normals at all the points on a given curve pass through a fixed point $\left(a,b\right)$. If the curve passes through $\left(3,-3\right)$and $\left(4,-2\sqrt{2}\right)$, given that $a-2\sqrt{2}b=3$, then…

Let $AD$ and $BC$be two vertical poles at $A\text{and}B$respectively on a horizontal ground. If $AD=8m$, $\mathrm{BC}=11m$, $\mathrm{AB}=10m;$ then the distance (in meters) of a point $M$lying in between $\mathrm{AB}$…

Let $f:\left[0,1\right]\rightarrow\,R$ be a twice differentiable function in $\left(0,1\right)$such that $f\left(0\right)=3$and $f\left(1\right)=5$. If the line $y=2x+3$intersects the graph of $f$at only two distinct…

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

Let $\mathrm{A}$ be the region enclosed by the parabola $y^2=2 x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $\mathrm{A}$ 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…