Finding the exact range of a function involving exponential and quadratic terms

[studymma168 (Wen Dao)]

FunctionRange[{y^2/4 + Sqrt[(x - y^2/4)^2 + (E^x - y)^2] + 1}, {x,
y}, t] // Quiet

Run the code above to get this range:

t >= 1.41421

Running the code below does not yield any results.

Minimize[{y^2/4 + Sqrt[(x - y^2/4)^2 + (E^x - y)^2] + 
   1, {x, y} \[Element] Reals}, {x, y}]

How to Find the Exact Range of a Function Involving Exponential and Quadratic Terms

The manual calculation process is as follows:

[mariuszi28cg (Mariusz Iwaniuk)]

I have only:

Open in Wolfram Cloud ↗

[gianlucagorni]

This shows that the function has only one stationary point, which is a local minimum, with a value of Sqrt[2]:

f = y^2/4 + Sqrt[(x - y^2/4)^2 + (E^x - y)^2] + 1;
grad = D[f, {{x, y}}];
stationary = Solve[grad == 0, {x, y}, Reals]
f /. stationary[[1]] // FullSimplify
hess = D[grad, {{x, y}}] /. stationary[[1]] // FullSimplify;
Eigenvalues[hess] // N