Find a local minimum of the function f(x) = x^3 - 2x.
To find a local minimum (i.e. a point where the function changes from a negative slope into a positive slope), we first need to find all points where the slope of the function is zero. The first derivative of a function gives information about the slope, so we find the first derivative ( f'(x) = 3x2 - 2 ) and set it equal to zero. 0 = 3x2 - 2 can be rearranged to x2 = 2/3. Taking the square root of both sides of the equation the slope is zero in the two points at approx. x = 0.816 and x = -0.816. In order to find out which of the two extrema is a minimum (not a maximum), we could simply check the slope just before and just after our two candidate points and confirm if the slope is indeed negative just before the extremum and positive just after. Alternatively we can find the rate of change of the slope in the two candidate points using the second derivative of f (f''(x) = 6x). For x = 0.816 the second derivative is f''(0.816) = 4.899. This positive rate of change indicates that the slope is increasing at this point. In other words, the slope at this moment is zero, but it is currently changing in a positive direction, so changing from a lower (a negative) value to a higher (a positive value). Finally we simply find the y-value by plugging x back into the original function --> f(0.816) = -1.09. We can conclude that the minimum can be found at (0.816 | -1.09).
