Help needed with Maths/Economics practice questions?

Have been revising maths for economics and have come across these four questions which I've been unable to solve correctly. Could anyone possibly help me out with some step-by-step solutions, I just want to know where I've gone wrong.

1) Suppose that a firm faces an inverse demand function p=50-4q relating price p to quantity q, and that the firm's total cost function is TC(q) = 16q^2+10q+6
a) Show that the firm's profit is always concave in q
b) Find the quantity or quantities which maximize profit locally.
c) Find the quantity of quantities which maximize profit globally.

2) A manufacturer has costs of production: C(q)=4q^2+5q+4
a) Derive expressions for the average cost and the marginal cost as a function of q.
b) Find the quantity which minimizes the average cost.
c) Verify that the marginal cost curve intersects the average cost curve at the lowest point of the average cost curve.

3) Suppose that a firm has costs of production given by C(q)=0.75q^2+8q+300
a) If the firm is a monopolist and the inverse demand function for its product is p=201 - 2q, find the most profitable output and price, q* and p*.
b) When maximising profit, does the firm produce at the minimum average cost?
c) If instead the firm sells in a perfectly competitive market where the price is p, what would be its most profitable output?

4) A bus company will charter a bus that holds 50 people to groups of 35 or more. If a group contains exactly 35 people, each person pays £60. In large groups, everybody's fare is reduced by £1 for each person in excess of 35. Determine the size of the group for which the bus company's revenue will be greatest.