Project ForgeEngineering Mathematics
Calculus
GATE 2020
Medium
Consider the functions: I. e^(−x), II. x² − sin(x), III. √(x³+1). Which of the above functions is/are increasing everywhere in [0,1]?
- III only
- BII only
- CII and III only
- DI and III only
Explanation
I: d/dx[e^(−x)] = −e^(−x) < 0 everywhere — strictly decreasing, not increasing. II: d/dx[x²−sin x] = 2x−cos x, which is −1 at x=0 (negative) but 2−cos(1)≈1.46 at x=1 (positive) — the derivative changes sign, so this function is NOT increasing everywhere on [0,1]. III: d/dx[√(x³+1)] = 3x²/(2√(x³+1)) ≥ 0 for all x≥0, zero only at the single point x=0, so the function is still strictly increasing across the whole interval. Only III qualifies.
Source: GATE CS 2020 (2020)
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