  Recent studies in one dimension have revealed that the temporal advantage
rendered by stochastic resetting to diffusing particles in attaining first
passage, may be annulled by a sufficiently strong attractive potential. We
extend the results to higher dimensions. For a diffusing particle in an
attractive potential $V({R})=k {R}^n$, in general $d$ dimensions, we study the
critical strength $k = k_c$ above which resetting becomes disadvantageous. The
point of continuous transition may be exactly found even in cases where the
problem with resetting is not solvable, provided the first two moments of the
problem without resetting are known. We find the dimensionless critical
strength $\kappa_{c,n}(k_c)$ exactly when $d/n$ and $2/n$ take positive
integral values. Also for the limiting case of a box potential (representing $n
\to \infty$), and the special case of a logarithmic potential $k
\ln\big(\frac{R}{a}\big)$, we find the corresponding transition points
$\kappa_{c,\infty}$ and $\kappa_{c,l}$ exactly for any dimension $d$. The
asymptotic forms of the critical strengths at large dimensions $d$ are
interesting. We show that for the power law potential, for any $n \in
(0,\infty)$, the dimensionless critical strength $\kappa_{c,n} \sim
d^{\frac{1}{n}}$ at large $d$. For the box potential, asymptotically,
$\kappa_{c,\infty} \sim (1 - \ln(\frac{d}{2})/d)$, while for the logarithmic
potential, $\kappa_{c,l} \sim d$.
