Final answer:
The correct increasing function that satisfies (f-1)'(20) = 1/5 is option D: f(x) = eˣ + 5x + 19.
Step-by-step explanation:
To find the increasing function that satisfies (f-1)'(20) = 1/5, we need to differentiate each function and evaluate it at x = 20.
If the derivative equals 1/5, then the function is increasing.
Let's consider each option:
A. f(x) = x + 5: The derivative is 1, which is not equal to 1/5.
B. f(x) = x³ + 5x + 20: The derivative is 3x² + 5, which is not equal to 1/5 when x = 20.
C. f(x) = x⁵ + 5x + 14: The derivative is 5x⁴ + 5, which is not equal to 1/5 when x = 20.
D. f(x) = eˣ + 5x + 19: The derivative is eˣ + 5, which is equal to 1 when x = ln(4) + 1.
This option satisfies the given condition, so the correct answer is D. f(x)= eˣ+ 5x + 19