Math & Data Science Q&A - Get answers from our AI that are verified by human experts
First time here? Checkout the FAQs!
x

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

0 like 0 dislike
87 views
Show that \(\sin (x) \leq x\) for \(x \geq 0\) and \(\sin (x) \geq x\) for \(x \leq 0\). Also show that \(\cos (x) \geq 1-\frac{1}{2} x^{2}\) for all \(x \in \mathbb{R}\).
in Mathematics by Diamond (71,587 points) | 87 views

1 Answer

0 like 0 dislike
Best answer

Proof.

Consider \(f(x)=\sin (x)-x\). Then \(f^{\prime}(x)=\cos (x)-1 \leq 0\). This means that \(f(x)\) is weakly decreasing on \(\mathbb{R}\). Since \(f(0)=0\), we have \(f(x) \leq 0\) for all \(x \geq 0\) and \(f(x) \geq 0\) for all \(x \leq 0\). Now consider \(g(x)=\cos (x)-1+\frac{1}{2} x^{2}\). we have \(g^{\prime}(x)=-\sin (x)+x=-f(x)\). Therefore \(g^{\prime}(x) \geq 0\) for \(x \geq 0\) and \(g^{\prime}(x) \leq 0\) for \(x \leq 0\). It follows that \(g\) is weakly increasing for \(x \geq 0\) and \(g\) is weakly decreasing for \(x \leq 0\). Since \(g(0)=0\) we have \(g(x) \geq 0\) for all \(x \in \mathbb{R}\)

by Diamond (71,587 points)

Related questions

0 like 0 dislike
1 answer
0 like 0 dislike
1 answer
asked Jun 16, 2022 in Mathematics by Gauss Diamond (71,587 points) | 71 views
0 like 0 dislike
1 answer

Join MathsGee, where you get expert-verified math and data science education support from our community fast.


On the MathsGee, you can:


1. Ask and answer questions


2. Get expert-verified answers


3. Vote on questions and answers


4. Tip your favorite community members


5. Join expert live video sessions (Paid/Free)


6. Earn points by participating


7. Start a Fundraiser



Posting on MathsGee


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




MathsGee Rules


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.




MathsGee Android Q&A

MathsGee Android Q&A