Question

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

• A. g(K) attains its maximum at the midpoint of (a,b)
• B. g(K) attains its minimum at two points in (a,b)
• C. g(K) attains its both maximum and minimum in (a,b)
• D. g(K) attains no maximum and no minimum in (a,b)

Hint:

For a quadratic to be positive everywhere, $A>0$ and $\Delta<0$. On an open interval, a strictly monotonic function does not achieve its extrema within the interval.

Answer 1

The Correct Option is D