Guess the Polynomial

There are infinitely many possible values of $f(5)$ . In fact, all the given options could be $f(5)$ :

$\begin{cases} f(x) = \dfrac{25}{6}x^4 -\dfrac{125}{3}x^3+\dfrac{881}{6}x^2 - \dfrac{625}{3}x + 100&\quad f(5) = 125\\ f(x) = 25x^4-250x^3+876x^2-1250x+600 &\quad f(5)=625\\ f(x) = -\dfrac{3}{8}x^4 +\dfrac{15}{4}x^3-\dfrac{97}{8}x^2+\dfrac{75}{4}x-9 &\quad f(5)=16\\ f(x)=x^2 &\quad f(5) = 25\\ f(x) = -\dfrac{25}{24}x^4 +\dfrac{125}{12}x^3-\dfrac{851}{24}x^2+\dfrac{625}{12}x-25 &\quad f(5) = 0\\ f(x) = -x^4+10x^3-34x^2+50x-24 &\quad f(5) = 1\\ f(x) = \dfrac{11}{24}x^4 - \dfrac{55}{12}x^3 + \dfrac{409}{24}x^2 - \dfrac{275}{12}x + 11 &\quad f(5) = 36\\ \end{cases}$

I can list other values for $f(5)$ , or higher degree polynomials for the multiple choice options, but you get the picture. We do not have sufficient information, therefore $f(5)\;\boxed{\text{Cannot be Determined}}$

$\text{Data is insufficient ; the degree of the polynomial must be stated , because without it we will not be able to frame the polynomial.}$

Hence ,

$\color{#3D99F6}{"Cannot \ be \ Determined"}$

The answer varies with degree. If degree is 5, then infinite possibilites will be there, because we are given 4 equations and degree 5 needs 5 equation. So coefficients cannot be determined and hence infinite values of f(5) are there. Here I gave example and not only degree 5 but many infinite degrees have infininite answers so answer is obviously cannot be determined