Problem of Optimisation: A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible.

First, draw a diagram showing the rectangle, the circle and the unknowns . Then find the equations representing this problem (equation of a circle of center 0, x² +y²=4² (1) and of the area of the rectangle A=2x*2y (2)). Substitute one variable of eq(1) in eq (2) ==> A= 4xROOT(16-x²) (3).The largest area can be found by differentiating eq(3) (to find the local maximum of the equation). dA/dx = (64-8x²)/ROOT(16-x²) (using the formula for the differentiation of the product of two functions).The stationary points are the points for which dA/dx = 0. We compute and find x=ROOT(8) or x= - ROOT(8).To verify it is a maximum, we can take the second derivative of this point (and it must be less than 0) or plug the value just lower and higher than ROOT(8) in the first derivative to see if the function is increasing or decreasing at those points. Finally, determine the area by plugging ROOT(8) in (2), which gives A=32 sq inches