Class 12

Math

Calculus

Application of Derivatives

Find the maximum are of the isosceles triangle inscribed in the ellipse $a_{2}x_{2} +b_{2}y_{2} =1,$with its vertex at one end of major axis.

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Prove that the function f given by $f(x)=x_{2}−x+1$is neither strictly increasing nor strictly decreasing on $(1,1)$.

Find the maximum area of an isosceles triangle inscribed in the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$with its vertex at one end of the major axis.

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Find the equation of all lines having slope 2 and being tangent to the curve $y+x−32 =0$.

The line $y=x+1$is a tangent to the curve $y_{2}=4x$at the point(A) $(1,2)$ (B)$(2,1)$ (C) $(1,2)$ (D) $(1,2)$

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.

Show that the function f given by $f(x)=x_{3}−3x_{2}+4x,x∈R$is strictly increasing on R.

Find the absolute maximum and minimum values of the function f given by $f(x)=cos_{2}x+sinx,$$x∈[0,π]$