Hi, Say we have [itex]f(x) = sin(x)[/itex], then [itex]f'(x) = cos(x)[/itex] and [itex]f''(x) = -sin(x)[/itex] Let [itex]x = π/2[/itex], the slope at that point is zero: [itex]cos(x) = 0[/itex]. The second derivative gives, [itex]-sin(x) = -1[/itex]: that means the slope of the first derivative at point x is negative. Thus for the next closest value to x, the slope of the original function is negative. I want to know why the next closest value, say t, to any x has f(t)>f(x) if the slope at point x is positive, and f(t)<f(x) if the slope at point x is negative? Thanks for help.