Let f : R → R satisfy f ( x + y ) = f ( x ) f ( y ) , f ( 0 )  = 0 , f , ( 0 ) = 1 , and let g ( x ) = 3 x / f ( x ) . Which are true?

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Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMaxima-MinimaHard2 minQB

MathematicsHardmultiple choice

Let $f : R \to R$ satisfy $f (x + y) = f (x) f (y), f (0) \neq = 0, f, (0) = 1$ , and let $g (x) = 3 x / f (x)$ . Which are true?

Options:(select one or more)

Answer:

A, D

Solution:

The functional equation and derivative condition give $f (x) = e^{x}$ , so $g = 3 x e^{- x}$ . It has a maximum at $x = 1$ , no extremum at 0, and horizontal levels in $(0, 3/ e)$ cut the positive branch twice. Direct differentiation also verifies (A).

Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Maxima-Minima

⏱ 2mℹ️ Source: QB

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