(1)

It seems false the way you have stated the problem. For consider (ix + n). It can easily be shown that

Int(a,b) (ix + n)^2 dx = 0

for

[a,b] = [-sqrt(3n^2), sqrt(3n^2)]

But with (g(x) = x) then

Int(a,b) x(ix + n) dx = 2in^3 * sqrt(3)


(2)

Center the circle on the origin and use a simple max value formula:

(d/dx) chord length = 0 or undefined

(d/dx) 2 * sqrt(r^2 - x^2) = 0 or undefined

-2x * (1 / sqrt(r^2 - x^2)) = 0 or undefined

x = -r, 0, r

Plug in the values

2 * sqrt(r^2 - (-r)^2) = 0

2 * sqrt(r^2 - 0) = 2*r

2 * sqrt(r^2 - r^2) = 0

So, the max value is obviously the line (x = 0) which passes through origin, a.k.a the circle's center.

And finally:

Chord that passes through center of circle = Diameter

Done-and-done.