$f(x)=x^2-2(4 \mathrm{~K}-1) x+g(\mathrm{~K})>0 \forall x \in \mathbb{R}$ and for $\mathrm{K} \in(\mathrm{a}, \mathrm{b})$. If $g(\mathrm{~K})=15 \mathrm{~K}^2-2 \mathrm{~K}-7$, then

• A

  $g(\mathrm{~K})$ attains its maximum at the midpoint of $(\mathrm{a}, \mathrm{b})$
• B

  $g(\mathrm{~K})$ attains its minimum at two points in $(\mathrm{a}, \mathrm{b})$
• C

  $g(\mathrm{~K})$ attains its both maximum and minimum in $(\mathrm{a}, \mathrm{b})$
• Correct answer
  D

  $g(\mathrm{~K})$ attains no maximum and no minimum in $(\mathrm{a}, \mathrm{b})$

Hence, the correct answer is option: 'D'.