Horizontal tangent/point of inflection problem

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To find the value of b for the function y=x^4 + bx^2 + 8x + 1, both the first and second derivatives must be considered. The first derivative should be set to zero to identify the horizontal tangent, while the second derivative must also equal zero to determine the point of inflection. By solving the second derivative equation for b and substituting it into the first derivative equation, the value of x can be found. This approach ensures that both conditions are satisfied simultaneously. Ultimately, the correct value of b can be determined through this method.
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Homework Statement



The function y=x(to the forth)+bx(squared)+8x+1 has a horizontal tangent and a point of inflection for the same value of x. What must be the value of b?

Homework Equations



I think you have to find the derivative, set it equal to 0 and solve for x

The Attempt at a Solution

 
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That's one thing you have to do alright. But the second derivative also has to be zero, doesn't it? I would suggest you solve the equation for the second derivative equal to zero for b and substitute it into the first equation. THEN solve for x.
 

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