Let $a, b, c > 0$ and $d\in\mathbb{N}$. Then the function $$ f \colon \mathbb{R} \to \mathbb{R}; x \mapsto x + a \exp\left(-b (x^d - c)^2\right) $$ admits a unique local minimum in $(0,\infty)$ (if $a$ and $b$ are not too small, I suppose). To my knowledge one cannot give a closed form expression for the value of this minimum, but I am wondering whether there exists a good estimate or upper bound for it. I would also be interested in the case where $d=1$, if that is easier to answer.
Here is an examplary plot for $a = 2, b = 10, c = 1, d = 3$ (made with Wolfram Alpha):
Since the minimum doesn't seem to be computable analytically, I don't know how to approach the problem. I would be grateful for any hint or known results for related problems.
The case $d=1$ seems to be easy. Let $x=y+c$ to make $$f(y)=y+c+a e^{-b y^2} \implies f'(y)=1-2 a b y e^{-b y^2}$$ Let $by^2=t$ to face the equation $$1-2 a \sqrt{b} e^{-t} \sqrt{t}=0\implies t=-\frac{1}{2} W_{-1}\left(-\frac{1}{2 a^2 b}\right)$$ where appears Lambert function. Then the minimum corresponds to $$x=c+ \sqrt{-\frac{W_{-1}\left(-\frac{1}{2 a^2 b}\right)}{2 b} }$$