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

To solve the given problem, we need to analyze the quadratic function \( f(x) = x^2 - 2(4K-1)x + g(K) \) and ensure it remains positive for all real numbers \( x \). This implies that the quadratic must have no real roots, which occurs when its discriminant is less than zero.

The discriminant (\( \Delta \)) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). Here, for \( f(x) \):

• \(a = 1\)
• \(b = -2(4K - 1) = -8K + 2\)
• \(c = g(K)\)

The discriminant of \( f(x) \) is calculated as follows:

\(\Delta = (-8K + 2)^2 - 4 \cdot 1 \cdot g(K)\)

We need \(\Delta < 0\), which simplifies to:

\((-8K + 2)^2 < 4 \cdot g(K)\)

Now substituting \( g(K) = 15K^2 - 2K - 7 \) into the inequality we get:

\(64K^2 - 32K + 4 < 60K^2 - 8K - 28\)

Simplifying, we have:

\(4K^2 - 24K + 32 > 0\)

This inequality describes the valid range for \( K \) such that the quadratic equation is positive for all \( x \). Solving this quadratic inequality for \( K \) involves finding the roots of the equality:

\(4K^2 - 24K + 32 = 0\)

Solving, we get:

\(K = \frac{24 \pm \sqrt{24^2 - 4 \cdot 4 \cdot 32}}{2 \cdot 4}\)

\(K = \frac{24 \pm \sqrt{576 - 512}}{8} = \frac{24 \pm \sqrt{64}}{8}\)

\(K = \frac{24 \pm 8}{8}\)

\(K = 4 \text{ or } K = 2\)

The inequality holds true for:

\((a, b) = (-\infty, 2) \cup (4, \infty)\)

Therefore, \( g(K) \) does not attain any maximum or minimum in the open interval \( (a, b) \) since both endpoints are not included in the possible range.

Hence, the correct option is:

g(K) attains no maximum and no minimum in (a,b)