Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMonotonicity-LMVTMedium2 minQB

MathematicsMediumsingle choice

Passage / Comprehension

If $f (x)$ is defined in $[a, b]$ and:

(i) $f (x)$ is continuous on $[a, b]$

(ii) $f (x)$ is derivable on $(a, b)$

(iii) $f (a) = f (b)$

Then $f$ has at least one point $c \in (a, b)$ such that $f^{'} (c) = 0$ . This is known as Rolle's theorem. This means if $f (x)$ is a polynomial, then between any two roots of $f (x) = 0$ , there always lies a root of $f^{'} (x) = 0$ .

If a function $f$ is such that its derivative $f^{'}$ is continuous on $[a, b]$ and derivable on $(a, b)$ then there exists a number 'c' between 'a' and 'b' such that $f (b) = f (a) + (b - a) f^{'} (a) + (b - a)^{2} K$ then $K =$

Options:

Answer:

Solution:

Apply Taylor's theorem with Lagrange's form of remainder about x = a, up to the second-order term: f(b) = f(a) + (b − a)f′(a) + ((b − a)²/2)f″(c) for some c ∈ (a, b), since f′ is continuous on [a, b] and derivable on (a, b). Comparing with f(b) = f(a) + (b − a)f′(a) + (b − a)²K gives K = f″(c)/2.

Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Monotonicity-LMVT

⏱ 2mℹ️ Source: QB

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