Class 12

Math

Calculus

Application of Derivatives

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is $3 2R $. Also find the maximum volume.

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Let $f:[2,7]0,∞ $ be a continuous and differentiable function. Then show that $(f(7)−f(2))3(f(7))_{2}+(f(2))_{2}+f(2)f(7) =5f_{2}(c)f_{prime}(c),$ where $c∈[2,7]˙$

If the tangent at any point $(4m_{2},8m_{2})$ of $x_{3}−y_{2}=0$ is a normal to the curve $x_{3}−y_{2}=0$ , then find the value of $m˙$

The latest edge of a regular hexagonal pyramid is $1cm˙$ If the volume is maximum, then find its height.

The tangent to the parabola $y=x_{2}$ has been drawn so that the abscissa $x_{0}$ of the point of tangency belongs to the interval [1,2]. Find $x_{0}$ for which the triangle bounded by the tangent, the axis of ordinates, and the straight line $y=x02$ has the greatest area.

Let $f(x)=−sin_{3}x+3sin_{2}x+5$ on $[0,2π ]$ . Find the local maximum and local minimum of $f(x)˙$

A curve is defined parametrically be equations $x=t_{2}andy=t_{3}$ . A variable pair of perpendicular lines through the origin $O$ meet the curve of $PandQ$ . If the locus of the point of intersection of the tangents at $PandQ$ is $ay_{2}=bx−1,$ then the value of $(a+b)$ is____

Find the minimum value of $(x_{1}−x_{2})_{2}+(20x_{1} −(17−x_{2})(x_{2}−13) )_{2}$ where $x_{1}∈R_{+},x_{2}∈(13,17)$.

Show that the tangent to the curve $3xy_{2}−2x_{2}y=1at(1,1)$ meets the curve again at the point $(−516 ,−201 )˙$