Question

What is the slope of the tangent line at a minimum of a smooth curve?

Answer 1

The slope is `0`.

Minima (the plural of 'minimum') of smooth curves occur at turning points, which by definition are also stationary points. These are called stationary because at these points, the gradient function is equal to `0` (so the function isn't "moving", i.e. it's stationary). If the gradient function is equal to `0`, then the slope of the tangent line at that point is also equal to `0`.

An easy example to picture is `y=x^2`. It has a minimum at the origin, and it is also tangent to the `x`-axis at that point (which is horizontal, i.e. a slope of `0`). This is because `dy/dx=2x` in this case, and when `x=0`, `dy/dx=0`.