Question
upstudy study bank question image url

Identify the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.) \[ y=x \sqrt{100-x^{2}} \] increasing decreasing

Ask by Floyd Fleming. in the United States
Mar 16,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The function \( y = x \sqrt{100 - x^2} \) is: - **Increasing** on the interval \((-5\sqrt{2}, \, 5\sqrt{2})\). - **Decreasing** on the intervals \((-10, \, -5\sqrt{2})\) and \((5\sqrt{2}, \, 10)\). In interval notation: - **Increasing**: \((-5\sqrt{2}, \, 5\sqrt{2})\) - **Decreasing**: \((-10, \, -5\sqrt{2}) \cup (5\sqrt{2}, \, 10)\)

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

The Deep Dive

To determine where the function \( y = x \sqrt{100 - x^2} \) is increasing or decreasing, we can find its derivative. The function is defined for \( -10 < x < 10 \). 1. Take the derivative using the product and chain rules: \[ y' = \sqrt{100 - x^2} + x \cdot \frac{d}{dx}(100 - x^2)^{1/2} \] After simplifying, we find that: \[ y' = \sqrt{100 - x^2} - \frac{x^2}{\sqrt{100 - x^2}} \] Setting \( y' = 0 \) to find critical points leads to \( 100 - 2x^2 = 0 \), meaning \( x = -5 \) or \( x = 5 \). 2. Analyzing intervals: - For \( x \in (-10, -5) \): Choose \( x = -6 \), \( y' > 0 \) (increasing). - For \( x \in (-5, 5) \): Choose \( x = 0 \), \( y' < 0 \) (decreasing). - For \( x \in (5, 10) \): Choose \( x = 6 \), \( y' > 0 \) (increasing). Thus, the function is: - Increasing on the intervals \( (-10, -5) \) and \( (5, 10) \). - Decreasing on the interval \( (-5, 5) \). To sum it up: Increasing: \( (-10, -5) \cup (5, 10) \) Decreasing: \( (-5, 5) \)

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy