Calculus, Local Maximum and Minimum

Local Maximum and Minimum, Critical Point

If f is a real valued function with f′(x) positive, f is increasing at x.  Let y = f(x) and let s (the slope) = f′(x).  Select ε = s/2, and determine h such that f(x+h) - f(x) over h is within ε of s.  Within this range, the quotient always exceeds s/2, hence it is positive.  The numerator has the same sign as h.  When we go forward f increases, and when we go backward f decreases.  Thus a positive derivative implies f is increasing, at least near x.  A negative derivative implies f is decreasing.

If the derivative is 0 the point is called a critical point.  It could be a maximum, or minimum, or neither, as shown by ±xn at the origin.  Either branch could be increasing or decreasing, depending on the sign prepended to x and the parity of the exponent.

As a corollary, f′(x) must equal 0 if f attains a local minimum or a local maximum at x.  A nonzero derivative means f is increasing or decreasing.  If we want to find the highest or lowest point on a curve, we only need look where the derivative is 0, the critical points.

If you graph x3-3x, you'll see a hump to the left of the y axis and a dip to the right.  Where exactly is the top of the hump and the bottom of the dip?  Differentiate to get 3x2-3 = 0, hence x = ±1.