Provide a mathematical proof for the general result that, given a linear average curve, the corresponding marginal curve must have the same vertical intercept but will be twice as steep as the average curve
That comment said, I can provide an answer here.
Take an average cost function:
$$f(x) = mx + b$$ where x is output, b is the intercept, and m is the gradient.
So total cost is:
$$C(x) = x(mx + b) = mx^2 + bx$$
And that makes marginal cost the derivative of this function:
$$C'(x) = 2mx + b$$
Best of luck with your homework.