Finding b in a Function with Horizontal Tangent and Point of Inflection

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SUMMARY

The function y=x^4+bx^2+8x+1 requires determining the value of b such that it has both a horizontal tangent and a point of inflection at the same x-coordinate. To find b, one must analyze the first and second derivatives of the function. The horizontal tangent condition is met when the first derivative equals zero, while the point of inflection is identified when the second derivative equals zero. The correct value of b can be derived from these conditions, leading to the conclusion that b must equal -6.

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  • Understanding of calculus concepts such as derivatives and points of inflection.
  • Familiarity with polynomial functions and their properties.
  • Knowledge of how to compute first and second derivatives.
  • Ability to solve equations involving derivatives.
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  • Study the process of finding points of inflection in polynomial functions.
  • Learn how to derive and analyze first and second derivatives of functions.
  • Explore the implications of horizontal tangents in calculus.
  • Practice similar problems involving polynomial functions and their derivatives.
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hamburgler
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I'm doing a calc BC practice exam and I found this question that stumps me to the max:

The function y=x^4+bx^2+8x+1 has a horizontal tangent and a point of inflection for the same value of x. What must be the value of b?


I know that points of inflection are found from the second derivative and setting that function to zero, however the 'b' value really throws me off

Oh btw here are the mc answers.

a) -6 b)-1 c) 1 d) 4 e) 6
 
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under what conditions does the function have a hotizontal tangent
Under what conditions does the function have a point of inflection
once you know that it should be easy to see when both occur
you will also see what b must be
 

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