Class 12

Math

Calculus

Application of Derivatives

Show that the function given by $f(x)=xgx $has maximum at $x=e$.

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Show that semi-vertical angle of right circular cone of given surface area and maximum volume is $sin_{−1}(31 )$.

The maximum value of $[x(x−1)+1]_{31},0≤x≤1$is(A) $(31 )_{31}$ (B) $21 $ (C) 1 (D) 0

Show that the altitude of a right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3..

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

Find the slope of the tangent to the curve $y=x_{3}−3x+2$ at the point whose x-coordinate is 3.

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:(i) $f(x)=x_{2},x∈[−2,2]$ (ii) $f(x)=sinx+cosx,x∈[0,π]$(iii) $f(x)=4x−21 x_{2},x∈[−2,29 ]$ (iv) f(x)=(x-1)

The total revenue in Rupees received from the sale of x units of a product is given by $R(x)=13x_{2}+26x+15$. Find the marginal revenue when$x=7$.

The length x of a rectangle is decreasing at the rate of 3 cm/minute and the width y is increasing at the rate of 2cm/minute. When $x=10$cm and $y=6$cm, find the rates of change of (a) the perimeter and (b) the area of the rectangle.