Suppose p,p' are local extremes. Reformulating the above computations these are characterised by parameters \vec{\lambda},\vec{\lambda}'\in\mathbb{R}^{n} via p(x)=\frac{e^{\langle\vec{\lambda},\vec{f}(x)\rangle}}{C(\vec{\lambda})} and similarly for p', where C(\vec{\lambda})=\int_{x\in\mathbb{R}} e^{\langle\vec{\lambda},\vec{f}(x)\rangle}~dx. We now note a series of identities: Via the satisfaction of the expectation-constraints and utilising gradients/directional derivatives, one has D\log(C(\cdot))\vert_{\vec{\lambda}}=\left.\frac{DC(\cdot)}{C(\cdot)}\right|_{\vec{\lambda}}=\mathbb{E}_{p}[\vec{f}(X)]=\vec{a} and similarly for \vec{\lambda}'. Letting u=\vec{\lambda}'-\vec{\lambda}\in\mathbb{R}^{n} one obtains:
