Determine the maximum area of a right triangle that has legs equal to x+1 and 2x+4.

To determine the maximum area of right triangle that has
the legs x+1 and 2x+4 .

The area A(x) of the right angled
triangles, whose right angle is formed of the sides of length x+1 and 2x+4 is given
by:

A(x) =
(1/2)(x+1)(2x+4).

A(x) =
(1/2)(2x^2+6x+4).

A(x) is maximum if A'(c) = 0, for which
A"(c) < 0.

A'(x) =
(1/2))(2x^2+6x+4)'

A'(x) =
(1/2){2*2*x+6}.

A'(x) = 0 gives  (1/2){4x+6} = 0, Or x = c=
-6/4 = -2/3.

A''(x) = (1/2)*4 >
0.

So A(x) is minimum when x = -2/3. So min A(x) = A(-2/3)
= (1/2)(-2/3+1)(2*-2/3+4) = (1/2) (1/3)(8/3)= 4/3.

As A(x)
= (1/2){2x^2+6x+4}, there is no maximum. A(x) increases unbounded for large x's (both
positive and negative).