Solving f'(x) = 0 at x=1 with a+b=0

  • Context: Undergrad 
  • Thread starter Thread starter Nguyen Thanh Nam
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Discussion Overview

The discussion revolves around finding values for parameters a and b in a piecewise function defined as f(x) = √(2 - x²) for -√2 ≤ x ≤ 1 and f(x) = x² + ax + b for x > 1, such that the function is continuous and has a derivative at x = 1. The focus is on the conditions required for continuity and differentiability at the point x = 1.

Discussion Character

  • Homework-related

Main Points Raised

  • One participant states that for the function to be continuous at x = 1, the condition a + b = 0 must hold.
  • The same participant attempts to analyze the derivative from the right side (x approaching 1 from greater than 1) and notes that it equals 0.
  • The participant expresses difficulty in evaluating the left-hand limit of the derivative as x approaches 1 from less than 1.
  • Another participant questions the expectation for quick responses to the inquiry.
  • Subsequent posts include apologies for perceived impatience and misunderstandings regarding the forum dynamics.

Areas of Agreement / Disagreement

There is no clear consensus on the mathematical approach to finding the relationship between a and b, and the discussion remains unresolved regarding the derivative analysis.

Contextual Notes

The discussion lacks detailed exploration of the mathematical steps required to evaluate the left-hand limit of the derivative and how it relates to the conditions for continuity and differentiability.

Nguyen Thanh Nam
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Well, Sorry but I couldn't find help @ Homework section, I try out here:
Ok, they tell me to find a and b so that the function:
f(x)= *Root(2-x^2) if -root(2)<=x<=1
*x^2 + ax + b if x>1
has derivative at 1
I got that the condition for this graph to be continuous at 1 is a+b=0
And I moved to check out the derivative stuff:
When x->1+, the derivative is 0
But when 1->1-, igot stuck:
lim (x->1+) of [f(x)-f(1)]/(x-1) = lim (x->1+) of [x^2+ax+b-1]/(x-1)] (from the function at my first post, I got that when x>1, f(x)=x^2+ax+b) How can I figure out the relation between a and b so that I can put it and the one above to an equation?
Help me please, thanks!
 
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Sorry, mein Fehler
 
Very sorry. I though there are many guys here in the forums. sorry a lot!
 

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