Class 12

Math

Calculus

Application of Derivatives

Find points on the curve $9x_{2} +16y_{2} =1$at which the tangents are(i) parallel to x-axis (ii) parallel to y-axis.

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If the tangent to the curve $xy+ax+by=0$ at $(1,1)$ is inclined at an angle $tan_{−1}2$ with x-axis, then find $aandb?$

How many roots of the equation \displaystyle{\left({x}-{1}\right)}{\left({x}-{2}\right)}{\left({x}-{3}\right)}+{\left({x}-{1}\right)}{\left({x}-{2}\right)}{\left({x}-{4}\right)}+{\left({x}-{2}\right)}{\left({x}-{3}\right)}{\left({x}-{4}\right)}+{\left({x}-{1}\right)}{\left({x}-{3}\right)}{\left({x}-{4}\right)}={0} are positive?

$f(x)=[x]$ is step-up function. Is it a monotonically increasing function for $x∈R?$

Find the point on the curve $3x_{2}−4y_{2}=72$ which is nearest to the line $3x+2y+1=0.$

If the curve $C$ in the $xy$ plane has the equation $x_{2}+xy+y_{2}=1,$ then the fourth power of the greatest distance of a point on $C$ from the origin is___.

Find the equation of tangent and normal to the curve $x=(1+t_{2})2at_{2} ,y=(1+t_{2})2at_{3} $ at the point for which $t=21 ˙$

Find the value of $a$ if the curves $a_{2}x_{2} +4y_{2} =1andy_{3}=16x$ cut orthogonally.

Using Lagranges mean value theorem, prove that $bb−a <g(ab )<ab−a =a,$where $0<a<b˙$