Answer:
f(7) ≥ 19
Step-by-step explanation:To find the smallest possible value of f(7), we can use the Mean Value Theorem for Derivatives. According to this theorem, if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that:
f'(c) = (f(b) - f(a))/(b - a)
In this case, we know that f(2) = 14 and f'(x) ≥ 1 for 2 ≤ x ≤ 7. Therefore, we can apply the Mean Value Theorem to the interval [2, 7] to get:
f'(c) = (f(7) - f(2))/(7 - 2)
Since f'(x) ≥ 1 for 2 ≤ x ≤ 7, we have:
1 ≤ f'(c) = (f(7) - 14)/5
Multiplying both sides by 5 and adding 14, we get:
f(7) ≥ 19