Final answer:
The smallest possible value of f(4) is 16, calculated by taking the initial value f(1)=10 and adding the minimum possible increase based on the derivative f'(x)≥2 over the interval from x=1 to x=4.
Step-by-step explanation:
The question asks about the smallest possible value of a function f(4) given that f(1)=10 and f'(x)≥2 for 1≤x≤4. Since the derivative f'(x) represents the rate of change of the function, and it is greater than or equal to 2, the function is increasing at a rate of at least 2 units per 1 unit increase in x over the interval from 1 to 4. To find the smallest possible value of f(4), we assume the minimum rate of increase, which is 2.
Starting with an initial value of f(1)=10, if we increase x from 1 to 4, which is an increase of 3 units, and multiply that increase by the minimum rate of change, we get that the smallest increase in f(x) would be 3 units times 2, which is 6. Therefore, the smallest possible value of f(4) is obtained by adding this increase to the initial value of f(1), leading to:
f(4) = f(1) + 3*2 = 10 + 6 = 16.