The function f(x) = kx^3+6kx^2 + 18x + 17 is increasing on R (set of…

The correct answer is Option (2) → \(k \in (0,3/2)\)

Note: The given answer is as per the official NTA Answer Key. However, more than one answer is possible: Option (2) and Option (3) are both correct, as explained below.

f is increasing on R when \(f'(x) \ge 0\) for every real x:

    \(f'(x) = 3kx^2 + 12kx + 18 = 3\left(kx^2 + 4kx + 6\right)\)

For k = 0, \(f'(x) = 18 \gt 0\), so f is increasing.

For k < 0, \(kx^2 + 4kx + 6\) becomes negative for large x, so f is not increasing on R.

For k > 0, the quadratic \(kx^2 + 4kx + 6\) stays \(\ge 0\) for all x exactly when its discriminant is \(\le 0\):

    \((4k)^2 - 4(k)(6) \le 0\)
    \(16k^2 - 24k \le 0\)
    \(8k(2k - 3) \le 0\)
    \(0 \le k \le \frac{3}{2}\)

So f is increasing on R exactly when \(0 \le k \le \frac{3}{2}\). (At \(k = \frac{3}{2}\), \(f'(x) = \frac{9}{2}(x+2)^2\), which is 0 only at x = −2.) No option includes the end values; every value of k inside \(\left(0, \frac{3}{2}\right)\) works.

Option (1) \(k \in (0,3)\) — Incorrect. It contains k = 2, for which \(f'(x) = 6x^2 + 24x + 18 = 6(x+1)(x+3)\), negative for \(-3 \lt x \lt -1\).

Option (2) \(k \in (0,3/2)\) — Correct. Every k in this interval makes \(f'(x) \gt 0\) for all x; it is the widest of the given intervals that works.

Option (3) \(k \in (1/2, 3/2)\) — Should also be correct, though not the answer given by NTA. This interval lies inside \(\left(0, \frac{3}{2}\right)\), so for every k in it f is increasing on R. The question asks for a condition (“if”) under which f is increasing, and this condition is also enough.

Option (4) \(k \in (1/2,5/2)\) — Incorrect. It also contains k = 2, for which f decreases on (−3, −1).

Advice for the exam: When more than one option is correct, mark the one that answers the question most completely, here the widest interval of k, \(\left(0, \frac{3}{2}\right)\). If NTA’s provisional answer key marks a different option, challenge it during the answer-key challenge window, giving proper reasoning. NTA does revise its key when a challenge is well justified, so a sound challenge can raise your marks.