Let x_0 be the real number such that e^ x_0 +x_0=0. For a given real number , define g(x)= 3 x e^x+3 x- e^x- x 3 (e^x+1 ) for all real numbers x. Then which one of the following statements is TRUE?

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Quantrex Academy · Accepted Answer

Correct answer: C.

aligned & e^ x_0 +x_0=0 \ & g(x)= 3 x e^x+3 x- e^x- x 3 (e^x+1 ) \ & g(x)=x- (e^x+x ) 3 (e^x+1 ) , g(x)+e^ x_0 =x+e^ x_0 - 3 ( e^x+x e^x+1 ) \ & g(x)+e^ x_0 = (x-x_0 )- 3 ( e^x+x e^x+1 ) \ & As very clear g (x_0 )+x_0 0 aligned aligned & _ x x_0 | g(x)+e^ x_0 x-x_0 | 0 0 = _ x x_0 | g^ (x) 1 |= |g^ (x_0 ) | \ & g^ (x)=1- 3 ( (e^x+1 ) (e^x+1 )- (e^x+x ) e^x (e^x+1 )^2 ) \ & g^ (x_0 )=1- 3 ( (e^ x_0 +1 )^2 (e^ x_0 +1 )^2 )=1- 3 \ & _ x x_0 | g(x)+e^ x_0 x-x_0 |= |g^ (x_0 ) |= |1- 3 | aligned

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