Question

For a triangle ABC with three sides a, b, c, if a = 4 and b = 3, then the value
of c which yields maximum area is:

Answer 1

Let the angle opposite side c be C
then area = (1/2)(4)(3)sinC
d(area)/dC = 6cos C = 0 for a max of area
6cos C = 0
cosC = 0
C = 90° , so we have a right-angled triangle,
You should recognize the common 3-4-5 right-angled triangle, if not ...

c^2 = 3^2 + 4^2 = 25
c = 5