Final answer:
To prove the existence of a point c in the interval [a, b] for which x = (c-a)(c-b)f(c), consider that the product of two continuous functions is continuous and then apply the Intermediate Value Theorem to show that there is a c satisfying h(c) = 0 for h(x) = (x-a)(x-b)f(x).
Step-by-step explanation:
The question is asking to show that there exists a point c in the interval [a, b] for a continuous function f such that x = (c-a)(c-b)f(c). Consider two functions, g(x) = (x-a)(x-b) and h(x) = (x-a)(x-b)f(x). Since f is continuous on the closed interval and g(x) is a polynomial (hence also continuous), h(x) will be continuous as well since it's the product of two continuous functions.
Notice that g(a) = g(b) = 0, and if f(a) and f(b) are not both zeros, we have h(a) and h(b) as zeros. If f(a) or f(b) is zero, then we already found our c because x = (c-a)(c-b)f(c) is zero. If not, by the Intermediate Value Theorem, there must be a point c in (a, b) where h(c) = 0 because h(x) will take on every value between h(a) and h(b).
Therefore, there exists such c that satisfies the condition.