Let f ( x ) = x 3 + x + 1 , g = f − 1 , and h ( x ) = xg ( x ) − ∫ 0 g ( x ) f ( t ) d t . Which statements are true?

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Mathematics - Definite Integration Question with Solution | TestHub

MathematicsDefinite IntegrationProperties of definite integrationMedium2 minQB

MathematicsMediummultiple choice

Let $f (x) = x^{3} + x + 1, g = f^{- 1}$ , and $h (x) = xg (x) - \int_{0}^{g (x)} f (t) d t$ . Which statements are true?

Options:(select one or more)

Answer:

A, C

Solution:

Differentiating cancels the $g'$ terms: $h' (x) = g (x)$ . Since $f (1) = 3, g (3) = 1$ , so $h' (3) = 1$ . Also $h^{''} (3) = g^{'} (3) =$ $1/ f' (1) = 1/4$ .

Stream:JEESubject:MathematicsTopic:Definite IntegrationSubtopic:Properties of definite integration

⏱ 2mℹ️ Source: QB

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