Class 12

Math

Calculus

Application of Derivatives

For the curve $y=f(x)$ prove that (lenght n or mal)^2/(lenght or tanght)^2

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In the curve $x_{a}y_{b}=K_{a+b}$ , prove that the potion of the tangent intercepted between the coordinate axes is divided at its points of contact into segments which are in a constant ratio. (All the constants being positive).

On the curve $x_{3}=12y,$ find the interval of values of $x$ for which the abscissa changes at a faster rate than the ordinate?

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 $aandb$ .

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?

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

Discuss the global maxima and global minima of $f(x)=tan_{−1}(g)_{e}x$ in$[3 1 ,3 ]$

Draw the graph of $f(x)=x_{2}−xx_{2}−5x+6 $

Prove that all the point on the curve $y=x+sinx $ at which the tangent is parallel to x-axis lie on parabola.