Let a , b , c be three real numbers such that a 0∀ x ∈ ( a , c ) then

By LMVT, b − a f ( b ) − f ( a ) 0 makes f ′ increasing). Cross-multiplying gives ( c − b ) f ( a ) + ( b − a ) f ( c ) > ( c − a ) f ( b ) .

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Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMonotonicity-LMVTMedium2 minQB

MathematicsMediumsingle choice

Let $a, b, c$ be three real numbers such that $a < b < c$ . Let $f (x)$ be continuous $\forall x \in [a, c]$ and differentiable $\forall x \in (a, c)$ . If $f^{''} (x) > 0\forall x \in (a, c)$ then

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Solution:

By LMVT, $\frac{f ( b ) - f ( a )}{b - a} < \frac{f ( c ) - f ( b )}{c - b}$ (since $f^{''} > 0$ makes $f^{'}$ increasing). Cross-multiplying gives $(c - b) f (a) + (b - a) f (c) > (c - a) f (b)$ .

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

⏱ 2mℹ️ Source: QB

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