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For a function [tex]f(x)[/tex], we know that [tex]f(17) = 62[/tex] and [tex]f'(17) = -2[/tex].

Which of the following options is correct?

A. [tex]f'(x) = -2x[/tex]
B. [tex]f(x) = 62x - 2[/tex]
C. [tex]f(x) = 62 - 2x[/tex]
D. [tex]f'(x) = 62[/tex]

Answer :

Final answer:

Using the given information, the correct function is option c) f(x) = 62 - 2x, which gives the derivative f'(x) = -2 that matches the conditions f(17) = 62 and f'(17) = -2. Correct answer is option C.

Explanation:

Let's use the given information to identify the correct function and derivative. We know that f(17) = 62 and the derivative of the function at that point, f'(17) = -2. If we look at option a), f'(x) = -2x, and substitute x with 17, we get f'(17) = -2(17) = -34, which does not match the given derivative of -2. Hence, option a) is incorrect.

Looking at option b), f(x) = 62x - 2, we see that the derivative, found by taking the coefficient of x, would be 62, which contradicts the given derivative f'(17) = -2. Therefore, option b) is also incorrect.

Option c), f(x) = 62 - 2x, upon differentiation gives f'(x) = -2. If we substitute x with 17, we confirm f'(17) = -2, which is correct as per the given information. Additionally, evaluating f(17) = 62 - 2(17) provides us with the expected f(17) = 62. Therefore, option c) is the correct function and derivative pair that corresponds to the given information.

For option d), f'(x) = 62, this suggests a constant slope regardless of x, which does not fit the requirement f'(17) = -2. Hence, option d) is incorrect.

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