Math help for calculus

Question

Suppose that a right triangle has legs of lengths 5 cm and 7 cm. (Note that the legs of a right triangle are the two sides that are not the hypotenuse.) A rectangle is inscribed in this right triangle so that two sides of the rectangle lie along the legs. Find the largest possible area of such a rectangle.

(a) Suppose that  f '(x) = √ x − 2 x2 for x > 0, and that f (1) = 1. Find  f (4).

(b) Suppose that  f '(x) = 2 1 + x2 and that f (0) = 3. Find  f (1).

Answer 1

sides of half the triangle's sides, so 2.5 x 3.5 = 8.75 square cm.7

You might just realize it's half the given values of 5 and 7

or, you can do it with the derivative of the Area = A = base times height

A=bh = (7-x)(5/7)x

A' = 5 - 10x/7 = 0

10x/7 =5

x = 5(7/10) = 7/2 = 3.5 cm

y=(5/7)x = (5/7)3.5 = 2.5 cm