Question

(1) for the function f , defined by f(x) = 4x^3 - 4x^2 -7x -2 , there exist a point c €]-1/2 ,2[ satisfying f'(c) = 0
(2) for all even integral values of n, lim (x+1)^-n as x tends to infinity exists.

Answer 1

1) $f'=12x^2-8x-7$

$f'=0\iff x_1=-\frac{1}{2}; x_2=\frac{7}{6}$

both points are in the interval [$-\frac{1}{2};2$ ]

2) $\lim\limits_{x\to \infty}(x+1)^{-n}=\left\{\begin{matrix} 0 & n>0 \\ 1 & n=0 \\ \infty& n<0 \end{matrix}\right.$

the limit doesn't exist for all negative n