How to find the largest area

Question

Find the triangle of largest area that has two sides along the positive coordinate axes if its hypotenuse is tangent to the graph of 𝑦=4𝑒−𝑥.

Answer 1

Base of triangle = x (x > 0)

Height of triangle = y = 4e^-x (y > 0)

Area of triangle = A = (1/2)xy = (1/2)x(4e^-x) = 2xe^-x

dA/dx = 2[e^-x -xe^-x] = 2e^-x(1 - x)

dA/dx = 0 when x = 1

If 0 < x < 1, then dA/dx > 0. So, A is increasing

If x > 1, then dA/dx < 0, So, A is decreasing

Relative and absolute max when x = 1.

The right triangle with right angle at (0,0) and with legs along the positive coordinate axes has horizontal leg of length 1 and vertical leg of length 4/e.