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------------------------------------------------ For a function \( f(x) \), we know that \( f(15) = 74 \) and \( f'(15) = -3 \). Estimate \( f(16) \), \( f(14) \), and \( f(18) \).

a) \( f(16) \approx 71 \), \( f(14) \approx 77 \), \( f(18) \approx 68 \)
b) \( f(16) \approx 71 \), \( f(14) \approx 77 \), \( f(18) \approx 80 \)
c) \( f(16) \approx 71 \), \( f(14) \approx 71 \), \( f(18) \approx 77 \)
d) \( f(16) \approx 77 \), \( f(14) \approx 71 \), \( f(18) \approx 80 \)

To estimate the value of the function f(x) for values near 15 using the given information that f(15) = 74 and f'(15) = -3, we use the concept of linear approximation. This method is based on the idea that if we know the value of a function and its derivative at a point, we can make an approximate prediction for the function's value at points close to the known point.

The basic formula for linear approximation is:

f(x) ≈ f(a) + f'(a) * (x - a)

Using this formula:

To estimate f(16), we calculate 74 + (-3) * (16 - 15), which equals 71.
To estimate f(14), we calculate 74 + (-3) * (14 - 15), which equals 77.
To estimate f(18), we calculate 74 + (-3) * (18 - 15), which equals 68.

The correct estimates based on this method are therefore: f(16) ≈ 71, f(14) ≈ 77, and f(18) ≈ 68, which corresponds to option (a).