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
b. the function is never decreasing.
b. find each local maximum, if there are any.
(type exact answers.)
a. the function has a local maximum value at one value of x. the maximum value is f()=
b. the function has a local maximum value at two values of x. in increasing order of x - value, the
maximum values are f()= and f()=
c. the function has a local maximum value at three values of x. in increasing order of x - value, the
maximum values are f()=, f()=, and f()=
d. there are no local maxima

Explanation:

Step1: Find the derivative of the function

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

Step2: Find the critical points

Set \(f^\prime(x)=0\), so \(10(\ln x + 1)=0\)
\(\ln x=- 1\), then \(x = e^{-1}=\frac{1}{e}\)
The domain of \(y = f(x)\) is \((0,+\infty)\)

Step3: Determine the intervals of increase and decrease

• For \(0\lt x\lt\frac{1}{e}\), let \(x=\frac{1}{e^{2}}\), then \(f^\prime(\frac{1}{e^{2}})=10(\ln\frac{1}{e^{2}}+1)=10(-2 + 1)=-10\lt0\)

So \(f(x)\) is decreasing on the interval \((0,\frac{1}{e})\)

• For \(x\gt\frac{1}{e}\), let \(x = 1\), then \(f^\prime(1)=10(\ln1+1)=10(0 + 1)=10\gt0\)

So \(f(x)\) is increasing on the interval \((\frac{1}{e},+\infty)\)

Step4: Find local extrema

Since \(f(x)\) changes from decreasing to increasing at \(x=\frac{1}{e}\)
\(f(\frac{1}{e})=10\times\frac{1}{e}\ln\frac{1}{e}=-\frac{10}{e}\)
This is a local minimum. There is no local maximum.

Answer:

a. The function \(f(x)\) is decreasing on \((0,\frac{1}{e})\) and increasing on \((\frac{1}{e},+\infty)\)
b. D. There are no local maxima