= Solution
Each Cartesian component of one link has zero <mean> and <variance> $b^2/3$, with zero cross-component <covariance>. By the multivariate <central limit theorem>, for large $N$ on the coil scale the vector <probability density> is
$$
\boxed{p(\mathbf r)=\left(\frac{3}{2\pi Nb^2}\right)^{3/2}
\exp\left(-\frac{3r^2}{2Nb^2}\right)}.
$$
It is normalized with respect to $d^3r$; the radial <probability density> is $4\pi r^2p(\mathbf r)$. The <Gaussian limit of the end-to-end distribution of a freely jointed chain> is an approximation, not the exact finite-$N$ law near $r=Nb$.
At prescribed vector separation the configurational <entropy> is $S(\mathbf r)=k_B\ln p(\mathbf r)+\text{constant}$. The <ideal chain> has no extension-dependent internal energy, so its <Helmholtz free energy> is
$$
A(\mathbf r)=A(0)+\frac{3k_BT r^2}{2Nb^2}.
$$
The externally required <force> is $\nabla_{\mathbf r}A$, while the chain's restoring <force> has the opposite sign. Thus
$$
\boxed{\mathbf f_{\rm restore}=-\frac{3k_BT}{Nb^2}\mathbf r,\qquad
k_{\rm spring}=\frac{3k_BT}{Nb^2},\qquad
\langle z\rangle=\frac{Nb^2}{3k_BT}f}.
$$
The last equation also follows by multiplying $p(\mathbf r)$ by the <Boltzmann factor> $e^{fz/(k_BT)}$ and completing the square. This <entropic spring> resists extension because fewer link configurations realize a stretched chain. At the same small applied <force>, increasing <temperature> reduces extension; at the same extension it increases the required <force>. Do not differentiate $-k_BT\ln[4\pi r^2p(\mathbf r)]$ to obtain the fixed-vector Hooke law, since that radial measure includes an additional geometric multiplicity. The low-force condition is $fb/(k_BT)\ll1$, equivalently $\langle z\rangle\ll Nb$.
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