Question

For which of the following decreasing functions f does (f⁻¹)'(10) = 1/8 ? a) f (x) = 2ˣ b) f (x) = e⁻ˣ c) f (x) = ln(x) d) f (x) = 1 / x²

Answer 1

The only decreasing function f for which (f⁻¹)'(10) = 1/8 is f(x) = e⁻ˣ.

Step-by-step explanation:

To find the decreasing functions f for which (f⁻¹)'(10) = 1/8, we can evaluate the derivatives of the given functions at x = 10 and find when the result is equal to 1/8. Let's evaluate the derivatives:

1. f(x) = 2ˣ: (f⁻¹)'(x) = (log₂(x))'. At x = 10, (log₂(10))' = 1/((10ln(2)) = 1/(10*0.693) = 1/6.93, which is not equal to 1/8. So option a) is not correct.
2. f(x) = e⁻ˣ: (f⁻¹)'(x) = (ln(x))'. At x = 10, (ln(10))' = 1/10, which is equal to 1/8. So option b) is correct.
3. f(x) = ln(x): (f⁻¹)'(x) = (exp(x))'. At x = 10, (exp(10))' = e⁡²ⁿ which is not equal to 1/8. So option c) is not correct.
4. f(x) = 1 / x²: (f⁻¹)'(x) = (-√(x))'. At x = 10, (-√(10))' = -1/(2√(10)) which is not equal to 1/8. So option d) is not correct.

Therefore, the only option for which (f⁻¹)'(10) = 1/8 is option b) f(x) = e⁻ˣ.