Home › Previous year questions › GATE 2021 ST › Question 36GATE 2021 ST – Question 36 Calculus · Single Variable Calculus · 2 marks · Multiple choice
Define f:ℝ→ℝ by f(x)=x³sin x if x=0 or x is irrational,and f(p/q)=1/q³ for nonzero reduced rational p/q,q∈ℕ. Which holds?
f is not continuous at zero. f is not differentiable at zero. f is differentiable at zero with derivative zero. f is differentiable at zero with derivative one. Practise this question in The GATE Grind →
Show answer and explanation Correct answer: (C) f is differentiable at zero with derivative zero.
Explanation At zero,f=0. Along irrational x,the quotient f(x)/x=x²sin x→0. For rational x=p/q,|p|≥1 gives 1/q≤|x|;hence |f(x)/x|=1/(q²|p|)≤1/q²≤x²→0. Both types give derivative zero. **C**.
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