How do you find maximum of function?

Answer 1

It depends on the function and the level of your mathematical skills. It also depends on whether you are looking for a global maximum or a local one. For example, a cubic equation [y = ax3 + bx2 + cd + d] has no global maximum but it will usually have a local one. This is also the case for any equation that is asymptotically +infinity somewhere in its domain.

If a function is twice differentiable over the domain in question, differentiate it once, set the resulting derivative equal to zero and solve for the coordinates of the stationary point. Next, differentiate it again and evaluate the value of the second derivative at the stationary point. If this derivative is negative, you have a local maximum at the stationary point. But be careful at the edges of the domain.

All this does not help if the function is not twice differentiable. Sometimes there are other ways. For example, let P(X = x) be the [probability distribution] function that X, the sum of the numbers on two dice, is x.

Then it can easily be shown that

P(2) = P(12) = 1/36

P(3) = P(11) = 2/36

P(4) = P(10) = 3/36

P(5) = P(9) = 4/36

P(6) = P(8) = 5/36

P(7) = = 6/36

and P(x) = 0 elsewhere.

P is not a continuous function and so cannot be differentiable, but the table above shows that the maximum of the function is at P(7).