Find the volume of the largest cone can be inscribed in a sphere of radius R.
Let r be the base radius x is the distance O the center of the sphere from the base and V the volume of the are
Height h of the cone =R+x
∴ V=13πr2h=π3(R2−x2)(R+x)
=π3(R2+R2x−Rx2−x2)
∴ dVdx=π3[R2−2Rx−3x2]
d2Vdx2=π3[−2R−6x]
For max or min VdVdx=0
∴ R2−2Rx−3x2=0
⇒(R+x)(x−3x)=0 2) x=−R,x3 but x≠−R
When x=R3d2Vdx2<0 V is max only when x=R3
∴ Max V=13π(R2−R29)(R+R3)=32πR381=827(43πR3)=827 (volume of sphere)