Class 12

Math

Calculus

Application of Derivatives

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is $34r $.

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The two curves $x_{3}−3xy_{2}+2=0$ and $3x_{2}y−y_{3}−2=0$

Find $c$ of Lagranges mean value theorem for the function $f(x)=3x_{2}+5x+7$ in the interval $[1,3]˙$

Find the equation of all possible normals to the parabola $x_{2}=4y$ drawn from the point $(1,2)˙$

For the curve $xy=c,$ prove that the portion of the tangent intercepted between the coordinate axes is bisected at the point of contact.

Two cyclists start from the junction of two perpendicular roads, there velocities being $3um/m∈$ and $4um/m∈$ , respectively. Find the rate at which the two cyclists separate.

If the equation of the tangent to the curve $y_{2}=ax_{3}+b$ at point $(2,3)isy=4x−5$ , then find the values of $aandb$ .

If$f(x)andg(x)$ be two function which are defined and differentiable for all $x≥x_{0}˙$ If $f(x_{0})=g(x_{0})andf_{prime}(x)>g_{prime}(x)$ for all $f>x_{0},$ then prove that $f(x)>g(x)$ for all $x>x_{0}˙$

If $a>b>0,$ with the aid of Lagranges mean value theorem, prove that $nb_{n−1}(a−b)1.$ $nb_{n−1}(a−b)>a_{n}−b_{n}>na_{n−1}(a−b),if0<n<1.$