Suppose I know my function G is infinitely differentiable on the closed interval [a,b] and that all derivatives of G (including G itself) vanish at b. For any z in [a,b], I have by the FTC that [tex] \int_z^b G'(w) dw = G(b) - G(z). [/tex] Or, switching limits, [tex] \int_b^z G'(w) dw = G(z) - G(b). [/tex] One can integrate by parts on the left-hand side and obtain (this is basically what the integral remainder form of Taylor's theorem tells you) [tex] G(z) - G(b) = G'(b)(z-b) + \int_b^z (z-w)G''(w)dw, [/tex] or [itex]G(z) = \int_b^z (z-w)G''(w)dw[/itex] when you drop the terms that are zero (i.e., G(b) and G'(b)). Now, I want to take absolute values of both sides and do some estimating. I get [tex] |G(z)| \leq \int_b^z |z-w||G''(w)|dw \leq C \int_b^z |z-w|dw, [/tex] since we are assuming G is [itex]C^\infty[/itex] and therefore has bounded second derivative on [itex][z,b][/itex]. But on this interval, w > z, so |z-w| = w-z, so we have [tex] |G(z)| \leq C \int_b^z (w-z) dw, [/tex] which evaluates to [tex] |G(z)| \leq - \frac 12 C (z-b)^2, [/tex] which is telling me that a positive quantity is less than, or equal to, a negative number! What on earth have I done wrong here?