# Difference between gradient descent and finding stationary points with calculus?

Take the example of trying to optimize a regression line:

$$y = b + mx$$ around some data.

Method #1

You can do this by getting the partial derivatives of the error function:

$$z = (1/2) \Sigma(f(x) - y)^2$$ with respect to b and m, and then setting these equations to zero to find the stationary points.

Method #2

Use the gradient descent algorithm to find a local minima.

Question:

Method one 'seems' superior. Why couldn't you find at all the stationary points along the x and y axis (z is the error going upwards) and just pick the minimum values for both x and y? Why can't you do this and avoid the iterative process associated with gradient descent?

Where am I going wrong with my intuition..?

There is no need for gradient descent in this case. Method 1 is great. You set the gradient equal to $0$ and solve for $m$ and $b$.