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If f(x) is differentiable on (a, b) and f(a) = 0 and there exists a real number k such that |f'(x) ≤ k |f (x)| on [a, b], then f(x) is
  • a)
    non zero,
  • b)
    zero,
  • c)
    discontinuous at x = a+b/2
  • d)
    Continuous but not differentiable,
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If f(x) is differentiable on (a, b) and f(a) = 0 and there exists a re...
Choose x> a such that k(x1 - a) < 1,
let α = sup | f(t) | on a < t < x1 then by Lagrange’s mean value theorem

where a < t< x ≤ x1
implies f(x) - f(a) = (x - a) f '(t)
[since f(a) = 0]
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Most Upvoted Answer
If f(x) is differentiable on (a, b) and f(a) = 0 and there exists a re...
Choose x> a such that k(x1 - a) < 1,
let α = sup | f(t) | on a < t < x1 then by Lagrange’s mean value theorem

where a < t< x ≤ x1
implies f(x) - f(a) = (x - a) f '(t)
[since f(a) = 0]
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If f(x) is differentiable on (a, b) and f(a) = 0 and there exists a real number k such that |f'(x) ≤ k |f (x)| on [a, b], then f(x) isa)non zero, b)zero, c)discontinuous at x = a+b/2d)Continuous but not differentiable,Correct answer is option 'A'. Can you explain this answer?
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