# Taylor's theorem for multi variable function, purpose of $t?$

In Nocedal and Wright's book on Numerical Optimization. On page 14 we have theorem 2.1 - Taylors theorem, equation 2.6 which is, I think an approximation for a multi variable function f which is supposed to be twice continuously differentiable.

From the answers to this question I understand how 2.4 and 2.5 was derived. But I don't see how we can get 2.6. It looks like 2.5 was integrated to get 2.6 but it's not clear what the limits would be. Also I think it should be an approximation there in 2.6 since it's only the taylor's expansion to 2nd order terms.

if $$p$$ was a vector of small magnitude, to cause a perturbation around $$x_o$$ for the function $$f$$, I'd imagine the equation should look like this,

So instead of $$x+tp$$ in the 3rd term, I would expect just $$x$$. Also if it's an equality sign between LHS and RHS, I would need to mention the extra terms of 3rd order and beyond.

So is 2.6 correct, if it is how can I derive it?

• Note that these are not approximations, they are equalities. So there are no "leftover small terms". The tradeoff is that the third term involves evaluating $f$ at a point which is probably different from $x$.
– MPW
Aug 22, 2019 at 17:39
• Yeah they evaluate f at $x+tp$ for "some" t, which I think means that you can find a t for which the equality is satisfied? Aug 22, 2019 at 18:07
• Yes, precisely. It’s an existence statement.
– MPW
Aug 22, 2019 at 18:35

By the Taylor approximation in one dimension we have for every $$C^2$$-function $$\gamma:\Bbb R\to \Bbb R$$ and every $$t_0,t\in\Bbb R$$ $$\gamma(t)=\gamma(t_0)+\gamma'(t_0)(t-t_0)+R_1(t)$$ for some remainder $$R_1(t)$$ with $$\lim_{t\to t_0}\frac{R_1(t)}{t-t_0}=0$$. The remainder can be expressed in the Lagrange form which gives $$\gamma(t)=\gamma(t_0)+\gamma'(t_0)(t-t_0)+\frac12\gamma''(\xi)(t-t_0)^2$$ for some $$\xi$$ between $$t$$ and $$t_0$$. Choosing $$t_0=0$$, $$t=1$$ and $$\gamma(t):=f(x+tp)$$ we thus have $$\gamma(1)=\gamma(0)+\gamma'(0)+\frac12\gamma''(\xi)$$ for some $$\xi\in]0,1[$$. But this is exactly $$f(x+p)=f(x)+\nabla f(x)^Tp+\frac12 p^T\nabla^2f(x+\xi p)p$$ (since $$\frac{\gamma(t+s)-\gamma(t)}{s}=\frac{f(x+tp+sp)-f(x+tp)}{s}\to f'(x+tp)(p)=\nabla f(x+tp)^Tp$$ for $$s\to 0$$ implies that $$\gamma'(t)=f(x+tp)^Tp$$ and especially $$\gamma'(0)=f(x)^Tp$$ and similar for the second derivative).