Question

a. find the open interval(s) on which the function is increasing and decreasing
b. identify the functions local and absolute extreme values, if any, saying where they occur.
f(x)=10x ln x
a. on what open interval(s), if any, is the function increasing? select the correct choice below and fill in any
answer boxes within your choice.
oa. the function f is increasing on the open interval(s)
(type your answer in interval notation. type exact answers. use a comma to separate answers
as needed.)
ob. the function is never increasing

Explanation:

Step1: Find the derivative of the function

Use the product rule \((uv)^\prime = u^\prime v+uv^\prime\), where \(u = 10x\) and \(v=\ln x\).
\(u^\prime=10\), \(v^\prime=\frac{1}{x}\)
\(f^\prime(x)=10\ln x + 10x\times\frac{1}{x}=10\ln x + 10\)

Step2: Find the critical points

Set \(f^\prime(x)=0\), so \(10\ln x+10 = 0\)
\(\ln x=- 1\)
Using the property \(y = \ln x\Leftrightarrow x = e^{y}\), we get \(x = e^{-1}=\frac{1}{e}\)

Step3: Determine the sign of the derivative in intervals

The domain of \(f(x)=10x\ln x\) is \((0,+\infty)\)

• For the interval \((0,\frac{1}{e})\), let \(x=\frac{1}{e^{2}}\) (\(\frac{1}{e^{2}}\in(0,\frac{1}{e})\))

\(f^\prime(\frac{1}{e^{2}})=10\ln(\frac{1}{e^{2}})+10=10(-2)+10=-10<0\)

• For the interval \((\frac{1}{e},+\infty)\), let \(x = 1\) (\(1\in(\frac{1}{e},+\infty)\))

\(f^\prime(1)=10\ln(1)+10=10\times0 + 10=10>0\)

Answer:

A. The function \(f\) is increasing on the open interval\((\frac{1}{e},+\infty)\)