Q. 88 - Calculus

Question

Prove that every quadratic function has exactly one local extremum.

Step-by-Step Solution

Answer

It is proded that every quadratic function has exactly one local extremum. 

Step 1. Given Information

We are given that every quadratic function has exactly one local extremum. 

Step 2. Proving the statement

Consider the quadratic function,

$f\left(x\right)=a{x}^2+bx+c\text{; where }a\ne 0\text{. }$

The objective is to prove that f has exactly one local extremum. The first derivative of the function is given by,

$$\begin{gathered} f\text{'}\left(x\right)=\frac{d}{dx}\left(a{x}^2+bx+c\right) \\ =2 a x+b \end{gathered}$$

For critical values,

$$\begin{gathered} f\text{'}\left(x\right)=0 \\ 2ax+b=0 \\ 2ax=-b \\ x=-\frac{b}{2a} \end{gathered}$$

As x has only one real value, The function has exactly one local extremum. Therefore, every quadratic function has exactly one local extremum.