Let f be a non constant polynomial of degree k and let g:$$R\to R$$ be a bounded continuous function

Which of the following statements are necessarily true? 

Correct option is D

​$$\textbf{Intermediate value theorem:}$$​​

The Intermediate Value Theorem (IVT) states that if a continuous function takes two values at some points

and the function is continuous between these points, then it must take any value between them at some point in that interval.

Let,

Now Let $$f(x)=x^k\\[10pt]and,h(x)=f(x)-g(x)\\[10pt]\text{if k=odd:}\\[10pt] \lim_{n\to-\infty}[x^k-g(x)] =-\infty\ (\because \text{g is bounded}) \cdots(1)\\[10pt]\lim_{n\to \infty}[x^k-g(x)]=\infty , \text{and h(x) is continuous in R } \cdots(2)\\[10pt]$$

Applying I.V.P we can say that h(x) acuires all values in R.

$$\implies \exist x_0\in R:h(x)=0 \ \text{if k is odd}\\[10pt]\implies \exist x_0 \in R:f(x_0)=g(x_0)\ \text{if k is odd.}\\[10pt]\textbf{Option D is correct.}$$

And if k is even then  both the limits in (1) and (2) will be $$\infty$$ .

and we will not be able to apply I.V.P so all other statements are not necessarily true.