objective function derivative

Question

Let's assume I have an objective function:

$f(x) = x^2-b$

If I find partial derivative w.r.t $b$, I get:

$\frac{\partial f }{\partial b} = -1$.

But if I want to minimize it, I would set it to zero:

$0=1$

Now, isn't this bizarre? How can zero be one? How to interpret this?

Answer 1

This is natural as $f$ is linear in $b$. There exist no local minima. Basically derivative is constant and function value goes on decreasing as you move in that direction. This is similar to $f(x) = x+3$, you cannot maximize or minimize the function w.r.t $x$ by setting derivative equal to zero.