Question

Let $R\to R$ is a objective function and $f\left(x\right)$ is a polynomial of degree $n$, then

• A. $n$ must be odd
• B. $f^{\prime \prime \prime }\left(x\right)$ can be a constant
• C. $f^{\prime }\left(x\right)=0$ cannot have more that $2$ solutions
• D. $y=f^{\prime }\left(x\right)$ where $f:R\to R$ must be many one and into function

Answer 1

Answer: (A), (B), (D)

The correct options are
A $n$ must be odd
B $f^{\prime \prime \prime }\left(x\right)$ can be a constant
D $y=f^{\prime }\left(x\right)$ where $f:R\to R$ must be many one and into function

Analyzing the domain and corresponding co domain we found that both are $R\to R$ which is covering the whole range of real numbers. So $n$ has to be an $odd$ number.

if $n=even$ then co domain belongs to only positive real numbers $R_{+}$.

Now if we differentiate an odd degree polynomial then we will bw getting a even degree polynomial whose codomain is not a whole real number range hence, a subset of $R$.

Hence $y=f^{\prime }\left(x\right)$ is many one and into function.

and then if we consider a value $n=3$

we will get,

$f\left(x\right)=a{x}^3+c$

$f^{\prime }\left(x\right)=3a^2$

$f^{\prime \prime }\left(x\right)=6ax$

$f^{\prime \prime \prime }\left(x\right)=6a$ which is a constant for every real a.