I am in a summer Econ class and have the following practice question. I'm at a loss. Can someone please explain this to me? I know that if I cannot figure this out, I won't be able to figure out the actual problems assigned.

Use the following equation to demonstate why a firm producing at the output level where MR = MC will also be able to maximize its total profit (i.e. be at the point where marginal profit is equal to zero).

P = 170 - 5Q
TC = 40 + 50Q + 5Q(squared)

If you'll graph the pertinent functions, you can get a visual, intuitive grasp of why profits hit their max when marginal revenue equals marginal cost.

Start by graphing the Revenue function. You're given the Price / Quantity relationship:

P = 170 - 5Q. Since Total revenue is just Price x Quantity...

Revenue = PQ = . Graph this Revenue curve, and also put the Total Cost curve onto the same graph.

Now, looking at those two curves, remember that Profit = Revenue minus Cost. This just means that for any given output level Q, the firm's Profit is the length of the vertical gap between the two curves.

Next, notice how that vertical gap changes as production increases (i.e. as you move left to right across the graph). For a while, the Revenue curve is rising faster than the Cost curve. That means that the gap is increasing, and thus Profit is increasing. But at some level of production (in this case, it'll be at Q = 6 units), the situation reverses, and the Cost curve begins rising at a steeper slope than the Revenue curve. This makes the overall profit--the spread between the curves--diminish. It's therefore easy to see that the production level Q = 6 is the 'sweet spot'--the production level at which the curve-spread is the greatest, and hence profits are at their max.

Here's the thrilling finale, where we tie it back in to the original question. Informally, the steepness of the Revenue curve is Marginal Revenue. Ditto for the Cost curve. Below Q = 6, the Revenue curve is rising faster than the Cost curve; i.e.

Marginal Revenue > Marginal Cost

Above Q = 6, MC > MR; that is, the cost curve is the steeper of the two, and profit (the gap between the curve) is shrinking.

But at the sweet spot of Q = 6, both curves have, momentarily, the same 'steepness', which is just another way of saying...

MR = MC.

That ought to point you in the right direction, and kick things off a bit for you. Best of luck!