Class 12

Math

Calculus

Application of Derivatives

Find the intervals in which the function f given by $f(x)=2x_{2}−3x$is(a) strictly increasing (b) strictly decreasing

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

Using Lagranges mean value theorem, prove that $∣cosa−cosb∣≤∣a−b∣˙$

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

Water is dropped at the rate of 2 $m_{3}$/s into a cone of semi-vertical angle is $45_{∘}$ . If the rate at which periphery of water surface changes when the height of the water in the cone is 2m is d. Then the value of 5d is _____ m/sec

Find the equation of the normal to the curve $y=∣∣ x_{2}−∣∣ x∣∣atx=−2.$

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

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

If the tangent at $(1,1)$ on $y_{2}=x(2−x)_{2}$ meets the curve again at $P,$ then find coordinates of $P˙$