If f ( x ) is continuous in [ a , b ] and differentiable in ( a , b ) then 3 c 2 f ′ ( c ) = b 3 − a 3 f ( b ) − f ( a ) holds for

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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 $f (x)$ is continuous in $[a, b]$ and differentiable in $(a, b)$ then $\frac{f ^{'} ( c )}{3 c ^{2}} = \frac{f ( b ) - f ( a )}{b ^{3} - a ^{3}}$ holds for

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Answer:

Solution:

This is Cauchy's mean value theorem applied to f(x) and g(x) = x³ on [a, b]: there exists at least one c ∈ (a, b) such that (f(b) − f(a))/(g(b) − g(a)) = f′(c)/g′(c), i.e. (f(b) − f(a))/(b³ − a³) = f′(c)/(3c²). So the given relation holds for at least one c ∈ (a, b).

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

⏱ 2mℹ️ Source: QB

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