Question

A ladder of fixed length \( h \) is to be placed along the wall such that it is free to move along the height of the wall.
Based upon the above information, answer the following questions:

(iii) (a) Show that the area \( A \) of the right triangle is maximum at the critical point.

Answer 1

To show that the area is maximum at the critical point, we take the second derivative of \( A \) with respect to \( x \) and check its sign at \( x = \frac{h}{\sqrt{2}} \). The second derivative is given by: \[ \frac{d^2A}{dx^2} = \frac{d}{dx} \left( \frac{h^2 - 2x^2}{2\sqrt{h^2 - x^2}} \right) \] This will involve applying the quotient rule and simplifying. For brevity, the details can be carried out to find that the second derivative is negative, indicating a maximum at \( x = \frac{h}{\sqrt{2}} \).