Given:
[tex]f^{\prime}\left(x\right)=x-4,\text{ and}f\left(2\right)=-1[/tex]
To find:
The correct function.
Explanation:
Let us consider the function given in option D.
[tex]f(x)=\frac{x^2}{2}-4x+5[/tex]
Differentiating with respect to x we get,
[tex]\begin{gathered} f^{\prime}(x)=\frac{2x}{2}-4 \\ f^{\prime}(x)=x-4 \end{gathered}[/tex]
Substituting x = 2 in the function f(x), we get
[tex]\begin{gathered} f(2)=\frac{2^2}{2}-4(2)+5 \\ =2-8+5 \\ =-6+5 \\ f(2)=-1 \end{gathered}[/tex]
Therefore, the given conditions are satisfied.
So, the function is,
[tex]f(x)=\frac{x^{2}}{2}-4x+5[/tex]
Final answer: Option D
