Question

In this problem you will use Rolle's theorem to determine whether it is possible for the function
f(x) = 8 x^{7} + 7 x - 13
to have two or more real roots
(or, equivalently, whether the graph of y = f(x) crosses the x-axis two or more times).

Suppose that f(x) has at least two real roots. Choose two of these roots and call the smaller one a and the larger one b. By applying Rolle's theorem to f on the interval [a,b], there exists at least one number c in the interval (a,b) so that f'(c)= ____ . The values of the derivative f'(x)=_____ are always _____(changing/negative/zero/postive/undefined) , and therefore it is _____(plausible/unlikely/possible/impossible) for f(x) to have two or more real roots.

Answer 1

f'(x) = 56x^6 + 7
since no value of x can satisfy f'(x) = 0, there are no local max/min.

That is, since there is at least 1 real root (polynomial is of odd degree), and no turning points, there is exactly 1 real root.

Answer 2

thank youu :)