Why is second order derivative positive in minima and negative in maxima?

Ok let me give you an intuitive feeling about this
Imagine a differential curve with a local maxima (basically a peak).
Now can you imagine that (shown in figure) that the slope goes from positive to zero to negative moving right on the x axis. This means the change in the slope is negative
and the change in the slope wrt x is nothing but the second derivative
hence the second derivative at maxima is negative
Similar argument can be constructed at local minima