Q.When is a function $f(x)$ said to have a relative minimum at a point $x=c$?

Local Minimum (Definition)

Picture walking through a hilly landscape and arriving at the bottom of a small dip. You may not be at the lowest point of the whole region — there could be a deeper valley far away — but right here, every direction you step leads uphill. That dip is a local minimum: a point that is lower than everything immediately around it.

The precise definition

Let $f$ be a function defined on an interval, and let $c$ be a point in it. We say $f$ has a local minimum (also called a relative minimum) at $c$ if there is some $\delta > 0$ such that

In words: throughout a small neighbourhood $(c-\delta,\ c+\delta)$ around $c$, the value $f(c)$ is the smallest. The value $f(c)$ itself is called the local minimum value.

How it links to the derivative

If $f$ is differentiable at a local minimum $c$, the tangent there is horizontal, so

Such a point is called a critical point. But $f'(c) = 0$ on its own does not guarantee a minimum — it could be a maximum or a point of inflection. To confirm a minimum, check that the function stops falling and starts rising there:

• First-derivative test: $f'$ changes sign from negative to positive as $x$ increases through $c$ (going downhill, then uphill).
• Second-derivative test: if $f'(c) = 0$ and $f''(c) > 0$, then $c$ is a local minimum (the curve is concave up, like a bowl). …