Graded Quiz; Question 8

Hey I’m facing some trouble understanding the question here, the question states that f(x) is a positive real function and g(x) = log f(x). Just want some clarification over:

  1. What is a positive real function?
  2. I chose the answer “If x_{max} is a point where f(x_{max}) is a local minimum, then g(x_{max}) is also a local minimum” assuming that the answer is correct since the derivative of log would be \frac{1}{x}, but this was wrong.
  3. The question feedback did suggest that the answer in (2) would be true if we compose f with a decreasing function, but log isn’t? I do understand the terms compose, decreasing, but I just don’t understand this sentence as a whole.

Appreciate the help if anyone could shed some light over this


Hi @Khong_Jia_Ren!

Thanks for bringing your questions here! This makes us improve the content.

I will try to answer your queries, please feel free to reply.

  1. A positive real function is a function f such that f(x) > 0 for every x in its domain. This assumption is because \log(x) is defined only for positive values, so to compose \log with other function, that function must output only positive numbers.

  2. This answer is not present in the quiz. What you wrote is true, indeed.

  3. \log is not a decreasing function, it is increasing (its derivative is decreasing, though). Note that \frac{d\log}{dx} = \frac{1}{x} > 0 if x > 0, therefore \log is always increasing in its domain.

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