Problem : Find the critical points and inflection points of the function f (x) = x ^{4} -2x ^{2} (with domain the set of all real numbers). Which of the critical points are local minima? local maxima? Is there a global minimum or maximum?
We first calculate the derivatives of the function:
f'(x) | = | 4x ^{3} - 4x | |
= | 4(x + 1)x(x - 1) | ||
f''(x) | = | 12x ^{2} - 4 | |
= | 4(3x ^{2} - 1) |
f''(- 1) | = | 8 | |
f''(0) | = | -4 | |
f''(1) | = | 8 |
f (- 1) | = | -1 | |
f (0) | = | 0 | |
f (1) | = | -1 |
Problem : Find the inflection points of f (x) = e ^{-x2 } . (This famous function is called the gaussian.)
We compute the derivatives:
f'(x) | = | -2xe ^{-x2 }, | |
f''(x) | = | (- 2x)(- 2xe ^{-x2 }) + (- 2)(e ^{-x2 }) | |
= | (4x ^{2} -2)e ^{-x2 } |
Problem : Find a function f (x) with inflection points at x = 1 and x = 2 .
We need f''(1) = f''(2) = 0 , so we might as well let
f''(x) = (x - 1)(x - 2) = x ^{2} - 3x + 2 |
f'(x) = - + 2x |
f (x) = - + x ^{2} |