Home › UP Board Class 12 › mathematics › Application of Derivatives › Rolle's theorem requires that on [a, b], in addi…
Rolle's theorem requires that on [a, b], in addition to continuity + differentiability:
Af'(a) = f'(b) (equal derivatives at ends)
Bf''(a) = 0 = f''(b) (zero second derivative)
Cf(a) + f(b) = 0 (sum is zero)
Df(a) = f(b) (function takes same value at endpoints)
Answer & Solution
Correct answer: D. f(a) = f(b) (function takes same value at endpoints)
Rolle's theorem: f continuous on [a,b], differentiable on (a,b), AND f(a) = f(b). Conclusion: there exists c in (a,b) with f'(c) = 0. Mean Value Theorem is similar but doesn't require f(a) = f(b).
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