Question

How would a horizontal line work in the Extreme value theorem?

Answer 1

Hint: We explain the equation of the lines which are considered as horizontal lines. We form the equation and find the slope of the functions. We equate it with 0. Extremum points in a curve have slope value 0. We solve it to find the coordinates and the points.

Complete step by step answer:
We first explain the statement about horizontal lines. The horizontal lines will be parallel to the X-axis. They are perpendicular to the Y-axis.
The equation of any lines which are parallel to the X-axis can be defined as $y=c$. Here $c$ is a constant.
We need to find the relative extrema of the function $y=c$.
To find the extremum points we need to find the slope of the function and also the value of the point where the slope will be 0.
Extremum points in a curve have slope value 0.
The slope of the function $y=f\left( x \right)=c$ can be found from the derivative of the function ${{f}^{'}}\left( x \right)=\dfrac{d}{dx}\left[ f\left( x \right) \right]$.
The differentiation of a constant term is always 0.
We differentiate both sides of the function $y=c$ with respect to $x$.
$\begin{align}
  & y=c \\
 & \Rightarrow {{f}^{'}}\left( x \right)=\dfrac{d}{dx}\left[ c \right]=0 \\
\end{align}$.
Therefore, the curve $y=c$ has its extremum points at all points of the curve.
The reason being $\forall x\in \mathbb{R}$ the slope of the curve is 0.
All points of the curve $y=c$ has extremum value.

Note: We can also prove it from the graph of the curve $y=c$. This is a straight line. There is no sharp curve in the graph.