NCERT Class 12-Mathematics: Chapter –6 Application of Derivatives Part 5
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Question 14:
Find the area of greatest rectangle that can be inscribed in an ellipse
Answer:
Let be the rectangle of maximum area with sides and , where is a point on the ellipse as shown in the Fig.6.3. The area A of the rectangle is i.e. which gives (say)

Point on the Ellipse
Therefore,
Again,
Now,
At
Thus at is maximum and hence the area A is maximum.
Maximum area units.
Question 17:
Find the difference between the greatest and least values of the function , on
Answer:
Therefore,
Clearly, is the greatest value and is the least.
Therefore, difference
Question 18:
An isosceles triangle of vertical angle is inscribed in a circle of radius . Show that the area of triangle is maximum when
Answer:
Let ABC be an isosceles triangle inscribed in the circle with radius a such that .
(see fig. 16.4)

ABC be an Isosceles Triangle Inscribed in the Circle Radius
Therefore, area of the triangle ABC i.e.
Therefore,
Therefore,
Therefore, Area of triangle is maximum when