How do you know if the second derivative of an equation is a maximum or a minimum?
If the second derivative of an equation is positive (d2y /dx2 > 0), we can see that this point on a curve is a minimum. This is because, where the first derivative finds the gradient of a curve (how the slope changes with respect to a change in x), the second derivative finds how an increase in x of an incremental amount affects the change in the gradient. - if you imagine y=x2, for example, we know it looks like a U so has a minimum point - any increase in x from the minimum (move to the right) would lead to the gradient increasing, so is positive. (The first derivative of x2 is 2x, the second derivative is 2 (positive)).
Conversely it follows that if the second derivative is negative (d2y /dx2 < 0), the curve has a maximum because any increase in x of any tiny amount will lead to the gradient decreasing (getting more negative away from zero) - so the change in the gradient will be negative. For y = -x2 , which looks like an upside down U, the first derivative is -2x, and the second is -2 which shows there to be a maximum (which we know is true).
