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) = 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.
thank youu :)
Well, let's roll with it!
By applying Rolle's theorem, we know that if f(x) has at least two real roots, then there must be a number c in the interval (a,b) where f'(c) = 0.
Now, let's find the derivative of f(x). Taking the derivative of 8x^7 + 7x - 13, we get 56x^6 + 7.
The values of the derivative f'(x) = 56x^6 + 7 are always... well, they are always changing. It's like a rollercoaster ride!
Therefore, it is plausible for f(x) to have two or more real roots. Just like a clown car, it seems like this function has plenty of room for more roots!
To determine whether it is possible for the function f(x) = 8x^7 + 7x - 13 to have two or more real roots, we will apply Rolle's theorem.
Rolle's theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one number c in the interval (a, b) such that f'(c) = 0.
Suppose f(x) has at least two real roots. Let's call the smaller root a and the larger root 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) such that f'(c) = 0.
We need to find the derivative f'(x) of the function f(x) = 8x^7 + 7x - 13.
To do that, we differentiate each term of the function:
f'(x) = 8(7x^6) + 7(1) - 0
= 56x^6 + 7.
The values of the derivative f'(x) = 56x^6 + 7 are always changing, as the variable x is not fixed.
So, the values of f'(x) are always changing.
Therefore, it is plausible for f(x) to have two or more real roots.
To determine whether the function f(x) = 8x^7 + 7x - 13 has two or more real roots, we can apply Rolle's theorem.
Rolle's theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one number c in the interval (a, b) such that f'(c) = 0.
In this case, let's assume that f(x) has at least two real roots. We'll label the smaller root as 'a' and the larger root as 'b'.
By applying Rolle's theorem to f on the interval [a, b], we have that there exists at least one number c in the interval (a, b) such that f'(c) = 0.
To find f'(x), we need to differentiate f(x) with respect to x. Taking the derivative f'(x) gives:
f'(x) = 56x^6 + 7
Now that we have the derivative, we can examine its behavior to determine whether it can be zero within the interval (a, b).
The values of the derivative f'(x) = 56x^6 + 7 are always positive because the derivative of a polynomial with positive leading coefficient will be positive for all real values of x. Therefore, it is impossible for f(x) to have two or more real roots.
In conclusion, based on Rolle's theorem and the behavior of the derivative, it is impossible for the function f(x) = 8x^7 + 7x - 13 to have two or more real roots.