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Mathematics - Continuity - Differentiability Question with Solution | TestHub

MathematicsContinuity - DifferentiabilityContinuity- MiscellaneousHard2 minQB

MathematicsHardnumerical

A polynomial $f$ satisfies $f (x + y) = f (x) + f (y) + 3 x y (x + y) - 1$ for all real $x, y$ , and $f^{'} (0) = 1$ . Find $lim_{x \to \infty} \frac{2 f ( x )}{f ( 2 x )}$ .

Answer:

0.25

Solution:

Putting $x = y = 0$ gives $f (0) = 1$ . Differentiating in $y$ at 0 gives $f' (x) = 3 x^{2} + 1$ , hence $f (x) = x^{3} + x + 1$ . Therefore $2 f (x) / f (2 x) \to 2/8 = 1/4$ .

Stream:JEESubject:MathematicsTopic:Continuity - DifferentiabilitySubtopic:Continuity- Miscellaneous

⏱ 2mℹ️ Source: QB

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