Question
upstudy study bank question image url

Determine if the statement is true or false. If a statement is false, explain why. Suppose that \( f(x) \) is a polynomial, and that \( a \) and \( b \) are real numbers where \( a
Ask by Powell Clarke. in the United States
Mar 16,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The statement is **True**. If \( f(a) < 0 \) and \( f(b) < 0 \) for a continuous polynomial \( f(x) \) on the interval \( [a, b] \), then \( f(x) \) does not cross the x-axis within that interval, meaning it has no real zeros there.

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

False. The given statement claims that if \( f(a)<0 \) and \( f(b)<0 \), then \( f(x) \) has no real zeros on the interval \( [a, b] \). However, this is not necessarily true. A polynomial can dip below the x-axis and then return below it again without necessarily crossing the x-axis to create real zeros. For instance, the polynomial \( f(x) = (x-1)^2 - 1 \) is negative at both ends of the interval \( [0, 2] \) but never crosses the x-axis; thus, it has no real zeros in that interval. So, the correct conclusion would be that just having \( f(a)<0 \) and \( f(b)<0 \) does not guarantee the absence of real zeros between \( a \) and \( b \).

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