If f ( x ) = 3 x 2 + ∫ 0 x e − t f ( x − t ) d t , which statements are true?

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Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMaxima-MinimaHard2 minQB

MathematicsHardmultiple choice

If $f (x) = 3 x^{2} + \int_{0}^{x} e^{- t} f (x - t) d t$ , which statements are true?

Options:(select one or more)

Answer:

C, D

Solution:

Rewrite the convolution as $e^{- x} \int_{0}^{x} e^{t} f (t) d t$ . Differentiation gives $f' = 3 x^{2} + 6 x$ , and $f (0) = 0$ , hence $f = x^{3} + 3 x^{2}$ . Its graph gives two solutions of $f = 4$ , and direct integration on [-1, 1] gives 2.

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

⏱ 2mℹ️ Source: QB

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