Let x₀ be the real number such that e^ x₀ +x₀=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?

Step-by-step solution

aligned & e^ x₀ +x₀=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^ x₀ =x+e^ x₀ - 3 ( e^x+x e^x+1 ) & g(x)+e^ x₀ = (x-x₀ )- 3 ( e^x+x e^x+1 ) & As very clear g (x₀ )+x₀ 0 aligned aligned & _ x x₀ | g(x)+e^ x₀ x-x₀ | 0 0 = _ x x₀ | g^ (x) 1 |= |g^ (x₀ ) | & g^ (x)=1- 3 ( (e^x+1 ) (e^x+1 )- (e^x+x ) e^x (e^x+1 )^2 ) & g^ (x₀ )=1- 3 ( (e^ x₀ +1 )^2 (e^ x₀ +1 )^2 )=1- 3 & _ x x₀ | g(x)+e^ x₀ x-x₀ |= |g^ (x₀ ) |= |1- 3 | aligned