Final answer:
The equation cos(x) = x³ has at least one real solution because the cosine function and the cubic function intersect on their graphs.
Step-by-step explanation:
To prove that the equation cos(x) = x³ has at least one real solution, we can examine the behavior of the two functions. The cosine function, cos(x), oscillates between -1 and 1, while the cubic function, x³, increases or decreases without bound. By looking at the graphs of these functions, we can see that they intersect at least once. Therefore, the equation has at least one real solution.