The real valued function f(x) = x^15 +5x^9 + 10 is increasing for ___…

The correct answer is Option (4) → all real values of x

Note: The given answer is as per the official NTA Answer Key. However, more than one answer is possible: Option (1), Option (2), Option (3) and Option (4) are all correct, as explained below.

    \(f'(x) = 15x^{14} + 45x^{8}\)
    \(f'(x) = 15x^{8}(x^{6} + 3)\)

\(x^8 \ge 0\) and \(x^6 + 3 \gt 0\) for every real x, so \(f'(x) \ge 0\), and \(f'(x) = 0\) only at the single point x = 0. Hence f is strictly increasing on the whole real line, and so it is also increasing on every part of the real line.

Option (1) all positive real values of x — Should also be correct, though not the answer given by NTA. f increases for all real x, so it increases on \((0, \infty)\) too.

Option (2) all negative real values of x — Should also be correct, though not the answer given by NTA. \((-\infty, 0)\) is a part of the real line, so f increases there too.

Option (3) all non-positive real values of x — Should also be correct, though not the answer given by NTA. \((-\infty, 0]\) is a part of the real line, so f increases there too.

Option (4) all real values of x — Correct. \(f'(x) = 15x^{8}(x^{6} + 3) \ge 0\), zero only at x = 0, so f increases on the whole real line, the largest such set.

Advice for the exam: When more than one option is correct, mark the one that answers the question most completely, here all real values of x. 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.