Learning starts with a question
First time here? Checkout the FAQs!

*Math Image Search only works best with zoomed in and well cropped math screenshots. Check DEMO

0 like 0 dislike
Suppose \(f^{\prime}(x)>0\) in \((a, b)\). Prove that \(f\) is strictly increasing in \((a, b)\), and let \(g\) be its inverse function. Prove that \(g\) is differentiable, and that
g^{\prime}(f(x))=\frac{1}{f^{\prime}(x)} \quad(a<x<b)
in Mathematics by Diamond (66,769 points) | 71 views

1 Answer

0 like 0 dislike
Best answer
For any \(c, d\) with \(a<c<d<b\) there exists a point \(p \in(c, d)\) such that \(f(d)-f(c)=f^{\prime}(p)(d-c)>0\). Hence \(f(c)<f(d)\).

We know that the inverse function \(g\) is continuous. (Its restriction to each closed subinterval \([c, d]\) is continuous, and that is sufficient.) Now observe that if \(f(x)=y\) and \(f(x+h)=y+k\), we have
Since we know \(\lim \frac{1}{\varphi(t)}=\frac{1}{\lim \varphi(t)}\) provided \(\lim \varphi(t) \neq 0\), it follows that for any \(\varepsilon>0\) there exists \(\eta>0\) such that
if \(0<|h|<\eta\). Since \(h=g(y+k)-g(y)\), there exists \(\delta>0\) such that \(0<|h|<\eta\) if \(0<|k|<\delta\). The proof is now complete.
by Diamond (66,769 points)

Related questions

0 like 0 dislike
0 answers
0 like 0 dislike
0 answers
0 like 0 dislike
1 answer

Join the MathsGee Q&A forum where you get STEM education support to succeed from our community. Connect and Learn.

On the MathsGee Q&A Forum, you can:

1. Ask questions

2. Answer questions

3. Vote on questions and answers

4. Start a fundraiser

5. Tip your favorite community members

6. Create Live Video Tutorials (Paid/Free)

7. Join Live Video Tutorials (Paid/Free)

8. Earn points by participating

MathsGee Q&A forum post

1. Remember the human

2. Act like you would in real life

3. Find original source of content

4. Check for duplicates before publishing

5. Read the community guidelines


1. Answers to questions will be posted immediately after moderation

2. Questions will be queued for posting immediately after moderation

3. Depending on the number of messages we receive, you could wait up to 24 hours for your message to appear. But be patient as posts will appear after passing our moderation.


Acalytica | Web Analytics | SEO Reports | Social Proof Tool | Email Marketing

MathsGee Android Q&A