How to minimize given functional I confronted the next problem: we have certain values $\psi_1, \psi_2, \psi_3$ in 3 points $x_1, x_2, x_3$, also we have a general functtion with 2 undefined coefficients ($A,x_0$): $$\psi=A\exp(x-x_0)$$ We need to find this coefficients using the next functional: $$\Phi = (\psi(x_1)-\psi_1)^2+(\psi(x_2)-\psi_2)^2+(\psi(x_3)-\psi_3)^2)$$
The main goal is to compose the $\psi$ function which would be the best approximization for the given 3 values. 
As I undestand for finding $\psi(x_1), \psi(x_2), \psi (x_3)$ we need to minimize the functional $\Phi$. Can you provide us the best approach for such minimization or some other way for finding the coefficients of general function $\psi$. The photo with illustration is attached.

 A: So you have $\Phi(x_0, A)$, it is a smooth function and you need to find its minimum. Which methods do you know that would help you to do that? For example, taking derivative and equating it to zero?
A: You need and you can only determine one coefficient. The other can be chosen arbitrary.
$y=A\cdot e^{x_i-x_0}=A\cdot e^{x_i}\cdot e^{-x_0}$
$A$ and $e^{-x_0}$ are constants. The product can be replaced by $e^k$. This means that $A$ and $e^{-x_0}$ cannot be definetely specified. Now we have
$y=e^k\cdot e^{x_i}$ with $k\neq 0$
Taking logs
$ln(y)=k+x_i$
Thus the sum of squared residuals is
$ssr=\sum_{i=1}^n \left( ln(y_i)-k-x_i\right)^2 $
the derivative w.r.t k is
$\frac{\partial \ ssr}{\partial k}=\sum_{i=1}^n 2\cdot \left( ln(y_i)-k-x_i\right)\cdot (-1)=0$
$\sum_{i=1}^n  ln(y_i)-\sum_{i=1}^n k- \sum_{i=1}^n x_i=0$
$\sum_{i=1}^n  ln(y_i)-n\cdot k- n\cdot \overline x=0$
$n\cdot k=\sum_{i=1}^n  ln(y_i)-n\cdot \overline x$
$k=\frac{\sum_{i=1}^n  ln(y_i)}{n}-\overline x$
This k is $A\cdot e^{-x_0}$. As I said one of this factors can be chosen arbitrary. 
For example, if $k= 2$ then $x_0$ can be $-ln(2)$. Then the equation is 
$2=A\cdot e^{-(-ln(2))}=A\cdot e^{ln(2)}=A\cdot 2 \quad \Rightarrow \quad A=1 $
