Answer:
The function f(x) is f(x) = 4/5x^(7/2) + (17 - 4/5 * 4^(7/2)).
Explanation:
Given the information provided, we can determine the function f(x) using the given constraints:
f³(t) = 3/(t)^(1/2)
f(4) = 17
f²(4) = 4
To solve for f(x), let's start by integrating the first equation three times:
f''(t) = 3 ∫ (t)^(1/2) dt = 3t^(3/2) + C1
f'(t) = 3 ∫ (t)^(3/2) dt = 2t^(5/2) + C2
f(t) = 4/5t^(7/2) + C3
Now, let's use the given values of f(4) = 17 and f²(4) = 4 to solve for the constants C1, C2, and C3.
Substituting t = 4 into the expression for f(t), we get:
f(4) = 4/5 * 4^(7/2) + C3 = 17
Solving for C3, we get:
C3 = 17 - 4/5 * 4^(7/2)
Substituting t = 4 into the expression for f'(t), we get:
f'(4) = 2 * 4^(5/2) + C2 = 4 * 2^(5/2)
Solving for C2, we get:
C2 = 4 * 2^(5/2) - 2 * 4^(5/2) = 0
Therefore, the function f(x) is f(x) = 4/5x^(7/2) + (17 - 4/5 * 4^(7/2))