Find the critical numbers of the function

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Homework Help Overview

The discussion revolves around finding the critical numbers of the function y=√(1-x^2), which involves analyzing its derivative and identifying points where the derivative is zero or undefined.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the derivative of the function, y'=-x/√(1-x^2), and explore the implications of setting it to zero. There is confusion regarding the critical points derived from the equation, particularly when considering the removal of the square root. Questions arise about the validity of different approaches leading to seemingly conflicting answers.

Discussion Status

Some participants have noted the conditions for critical numbers and are attempting to clarify the specific values that meet these conditions. There is acknowledgment of multiple critical points, with a focus on the values 0, 1, and -1, although the exact interpretation of these points is still being explored.

Contextual Notes

Participants are considering the endpoints of the domain of the function and the conditions under which the derivative is undefined, which may influence the identification of critical numbers.

phillyolly
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Homework Statement



Find the critical numbers of the function
y=√(1-x^2)


The Attempt at a Solution



y'=-x/√(1-x^2 )=0
-x/√(1-x^2 )=0

Here, I stuck. If I continue with this equation, my answers are x1=0 and x2 is less than 1.
But if I try to get rid of the root in order to find the critical points by doubling the both sides of equation, my answer is x1=0, x2 not equal to 1.

These are quite different answers. Please help me out with the problem.
 
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phillyolly said:

Homework Statement



Find the critical numbers of the function
y=√(1-x^2)


The Attempt at a Solution



y'=-x/√(1-x^2 )=0
-x/√(1-x^2 )=0

Here, I stuck. If I continue with this equation, my answers are x1=0 and x2 is less than 1.
But if I try to get rid of the root in order to find the critical points by doubling the both sides of equation, my answer is x1=0, x2 not equal to 1.

These are quite different answers. Please help me out with the problem.
Critical numbers occur at places where the derivative is 0, or at places in the domain of the function where the derivative is undefined, or at endpoints of the domain.

Seems to me that there are three critical numbers.
 
Mark44 said:
Critical numbers occur at places where the derivative is 0, or at places in the domain of the function where the derivative is undefined, or at endpoints of the domain.

Seems to me that there are three critical numbers.

Do you mean 0, 1, and -1?

x1=0

1-x^2=0 =>x=+-1

Is that what you mean?
 

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