Asked by sheks
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.
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.