Class 12

Math

Calculus

Application of Derivatives

If in a triangle $ABC,$ the side $c$ and the angle $C$ remain constant, while the remaining elements are changed slightly, show that $cosAda +cosBdb =0.$

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Discuss the maxima and minima of the function $f(x)=x_{32}−x_{34}˙$ Draw the graph of $y=f(x)$ and find the range of $f(x)˙$

If $f(x)$ is continuous in $[a,b]$ and differentiable in (a,b), then prove that there exists at least one $c∈(a,b)$ such that $3c_{2}f_{prime}(c) =b_{3}−a_{3}f(b)−f(a) $

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

The tangent at any point on the curve $x=acos_{3}θ,y=asin_{3}θ$ meets the axes in $PandQ$ . Prove that the locus of the midpoint of $PQ$ is a circle.

Discuss the extremum of $f(x)=x(x_{2}−4)_{−31}$

For the curve $y=a1n(x_{2}−a_{2})$ , show that the sum of length of tangent and sub-tangent at any point is proportional to product of coordinates of point of tangency.

Discuss the extremum of $f(x)={∣∣ x_{2}−2∣∣ ,−1≤x<3 3 x ,3 ≤x<23 3−x,23 ≤x≤4$

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