If $ f(x)=x^{3} $ and $ g(x)=x^{3}-4 x $ in $ -2 \leq x \leq 2 $, then consider the statements: (a) $ f(x) $ and $ g(x) $ satisfy mean value theorem. (b) $ f(x) $ and $ g(x) $ both satisfy Rolle's theorem. (c) Only $ g(x) $ satisfies Rolle's theorem. Of these statements

Question

If $ f(x)=x^{3} $ and $ g(x)=x^{3}-4 x $ in $ -2 \leq x \leq 2 $, then consider the statements: (a) $ f(x) $ and $ g(x) $ satisfy mean value theorem. (b) $ f(x) $ and $ g(x) $ both satisfy Rolle's theorem. (c) Only $ g(x) $ satisfies Rolle's theorem. Of these statements, (a) alone is cortect, (a) and (c) are correct., (a) and (b) are correct., None is correct.

MARKS App · Accepted Answer

Given that, f ( x ) = x 3 and g ( x ) = x 3 − 4 x Since, f ( x ) and g ( x ) are both continuous at [ − 2 , 2 ] and differentiable at [ − 2 , 2 ] So, f ( x ) and g ( x ) satisfy mean value theorem. Now, f ( − 2 ) = − 8 , f ( 2 ) = 8 So, f ( − 2 )  = f ( 2 ) g ( 2 ) = ( 2 ) 3 − 4 ( 2 ) = 0 = g ( − 2 ) = ( − 2 ) 3 − 4 ( − 2 ) = 0 Therefore, f ( x ) does not satisfy Rolle's theorem but g ( x ) satisfy Rolle's theorem.

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