Final answer:
To show that the equation x^3 - 17x + c = 0 has at most one root in the interval [-2, 2], we analyze the derivative 3x^2 - 17. As the reasoning reveals, within the given interval, the quadratic derivative cannot be zero, ensuring the function is monotonic, and thus can intersect the x-axis at most once.
Step-by-step explanation:
The question involves showing that the cubic equation x^3 - 17x + c = 0 has at most one root in the interval [-2, 2]. We can utilize Rolle's theorem or the Intermediate Value Theorem for continuous functions to answer this question. If a continuous function has a root at two points within an interval, then there must be at least one point where the function's derivative is zero within that interval. The derivative of our function is 3x^2 - 17, which is a quadratic equation that can have at most two real roots.
By analyzing the sign changes of the derivative, we can see that 3x^2 - 17 = 0 has roots at x = -\sqrt{17/3} and x = \sqrt{17/3}. Since both these values are outside the interval [-2, 2], the derivative does not change sign within the interval which implies that the function x^3 - 17x + c is either strictly increasing or strictly decreasing within the interval. Therefore, by the Intermediate Value Theorem, if the cubic equation has a root within the interval, it can have at most one, because a continuous function that doesn't change its monotonicity can intersect the x-axis only once within an interval where its derivative does not change sign.