use rolle's theorem to show that the equation 7x^6-9x^2+2=0 has at least one solution in the interval (0,1)
let f(x) = 7x^6-9x^2+2
f(0) = 2
f(1) = 7-9+2 = 0 , well , isn't that special, we accidentally found a solution
So I guess there has to be at least one solution.
To apply Rolle's theorem, we need to check three conditions:
1. First, the function must be continuous on the closed interval [0, 1].
In this case, the equation is a polynomial function, and polynomials are continuous on the entire real number line. Therefore, it is also continuous on the interval [0, 1].
2. Next, the function must be differentiable on the open interval (0, 1).
The derivative of the equation 7x^6 - 9x^2 + 2 with respect to x is 42x^5 - 18x. This derivative exists and is defined for all values of x.
3. Finally, the function must have equal values at the endpoints of the interval.
The function f(x) = 7x^6 - 9x^2 + 2 evaluates to f(0) = 2 and f(1) = 0.
Since all three conditions of Rolle's theorem are satisfied, we can conclude that there exists at least one value in the interval (0,1) such that the derivative of the function is equal to zero. This means that the equation 7x^6 - 9x^2 + 2 = 0 has at least one solution in the interval (0,1).