Class 12

Math

Calculus

Application of Derivatives

Show that the function f given by $f(x)=tan_{−1}(sinx+cosx),$$x >0$is always an strictly increasing function in $(0,4π )$.

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Find the least value of a such that the function f given by $f(x)=x_{2}+ax+1$is strictly increasing on $(1,2)˙$

Prove that the function f given by $f(x)=gcosx$ is strictly decreasing on $(0,2π )$and strictly increasing on $(2π ,π)$

For the curve $y=4x_{3}−2x_{5},$find all the points at which the tangent passes through the origin.

Find the maximum and minimum values, if any, of the following functions given by (i) $f(x)=(2x−1)_{2}+3$ (ii) $f(x)=9x_{2}+12x+2$(iii) $f(x)=−(x−1)_{2}+10$ (iv) $g(x)=x_{3}+1$

Find the values of x for which $y=[x(x−2)]_{2}$is an increasing function

The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Find the rate of change of the area of a circle with respect to its radius r when(a) $r=3cm$ (b) $r=4cm$

Manufacturer can sell x items at a price of rupees $(5−100x )$each. The cost price of x items is Rs $(5x +500)$. Find the number of items he should sell to earn maximum profit