Let f(x)=3x^5+5x^3+15x+15.
Use Rolle's Theorem to show that f(x) has exactly one root.
f' = 15 x^4 + 15 x^2 + 15
= 15 (x^4+x^2+1)
where is that slope = 0?
x^2 = [ -1 +/- sqrt(1-4)]/ 2
x^2 = -1/2 +/- (1/2) sqrt (-3)
complex roots only, it never has zero slope so it can only cross the axis once by Rolle's theorem. Once it crosses the axis, it can never reverse and come back.
To show that the function f(x) has exactly one root, we can use Rolle's Theorem. Rolle's Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and if the function takes the same value at both endpoints, then there exists at least one point c in the interval (a, b) such that the derivative of the function is equal to zero at that point.
In this case, we have the function f(x) = 3x^5 + 5x^3 + 15x + 15. We need to show that there exists at least one point c where f'(c) = 0.
To find the derivative, f'(x), we need to find the derivative of each term of the function. The derivative of 3x^5 is 15x^4, the derivative of 5x^3 is 15x^2, and the derivative of 15x is 15. The derivative of the constant term 15 is 0, as the derivative of a constant is always zero. Adding all these derivatives, we have:
f'(x) = 15x^4 + 15x^2 + 15
Now, we need to find the values of x for which f'(x) = 0. We can set the derivative equal to zero and solve for x:
15x^4 + 15x^2 + 15 = 0
To simplify the equation, we can divide both sides by 15:
x^4 + x^2 + 1 = 0
This is a quadratic equation in terms of x^2. We can let y = x^2 and rewrite the equation as:
y^2 + y + 1 = 0
Now, we can solve this quadratic equation for y. However, when we solve this equation, we find that the discriminant is negative, which means that there are no real solutions for y. Since y = x^2, there are no real values of x for which f'(x) = 0.
As a result, we have shown that there is no point c in the interval (a, b) where the derivative f'(x) = 0. Therefore, by Rolle's Theorem, we can conclude that the function f(x) = 3x^5 + 5x^3 + 15x + 15 has exactly one root.