Class 12

Math

Calculus

Application of Derivatives

Find the intervals in which the function f given by $f(x)=x_{3}+x_{3}1 ,x=0$is (i) increasing (ii) decreasing.

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

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).

Find the angle between the curves $x_{2}−3y_{2} =a_{2}andC_{2}:xy_{3}=c$

If $1_{0}=α$ radians, then find the approximate value of $cos60_{0}1_{prime}˙$

Prove that $f(x)=x−sinx$ is an increasing function.

Determine $p$ such that the length of the such-tangent and sub-normal is equal for the curve $y=e_{px}+px$ at the point $(0,1)˙$

Let $P(x)$ be a polynomial with real coefficients, Let $a,b∈R,a<b,$ be two consecutive roots of $P(x)$ . Show that there exists $c$ such that $a≤c≤bandP_{prime}(c)+100P(c)=0.$

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___.