Mathematics - Application of Derivative Question with Solution | TestHub
MathematicsApplication of DerivativeMonotonicity-Rolle's-theroemMedium2 minQB
MathematicsMediumsingle choice
Passage / Comprehension
If is defined in and:
(i) is continuous on
(ii) is derivable on
(iii)
Then has at least one point such that . This is known as Rolle's theorem. This means if is a polynomial, then between any two roots of , there always lies a root of .
A twice differentiable function f such that and and for . Then there is atleast one value ' ' between a and b for which
Options:
Answer:
C
Solution:
Since f is continuous and f(a) = f(b) = 0 with f(c) > 0 (a < c < b), apply LMVT on [a, c]: there is c1 ∈ (a, c) with f′(c1) = f(c)/(c − a) > 0. Apply LMVT on [c, b]: there is c2 ∈ (c, b) with f′(c2) = −f(c)/(b − c) < 0. Now f′ is continuous on [c1, c2] and derivable on (c1, c2), so by LMVT again there is ε ∈ (c1, c2) ⊂ (a, b) with f″(ε) = (f′(c2) − f′(c1))/(c2 − c1) < 0.
Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Monotonicity-Rolle's-theroem
⏱ 2mℹ️ Source: QB
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