Q.4 - Calculus

Question

Describe what a critical point is, intuitively and in mathematical language. Then describe what a local extremum is. How are these two concepts related?

Step-by-Step Solution

Answer

The point at which the derivative vanishes or does not exist is known as the critical point.

That is a point $x=c$ is a critical point if $f\text{'}\left(c\right)=0$ or $f\text{'}\left(c\right)$ does not exist.

The local extremum is the point at which the function takes either minimum or maximum value.

If $x=c$ is a local extremum and $f$ is differentiable at $x=c$, then $f\text{'}\left(c\right)=0$.

Step 1. Given Information.

The function is f(x).

Step 2. Critical points.

The point at which the derivative vanishes or does not exist is known as the critical point.

That is a point $x=c$ is a critical point if $f\text{'}\left(c\right)=0$ or $f\text{'}\left(c\right)$ does not exist.

Every critical point need not be a local extremum of a function $f$.

Step 3. Local Extremum at a point.

The local extremum is the point at which the function takes either minimum or maximum value.

Suppose $x=c$ is the location of a critical point of a function $f$ with $f\text{'}\left(c\right)=0$, and suppose both $f$ and $f\text{'}$ are differentiable and $f\text{'}\text{'}$ is continuous on the interval around $x=c$.

If $x=c$is a local extremum and $f$is differentiable at $x=c$, then $f\text{'}\left(c\right)=0$.