Finding Local Extrema in Polynomial Equations

AI Thread Summary
To find the x-coordinates of local extrema in the given polynomial equations, one must first calculate the derivative of each function and set it to zero to identify critical points. The equations presented include cubic, quartic, and quintic polynomials, each requiring differentiation. After finding critical points, the second derivative test can be applied to determine whether these points are local maxima or minima. It is important to engage with the problem and show some effort to receive assistance, as per forum guidelines. Understanding these steps is crucial for successfully identifying local extrema in polynomial functions.
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how can i find the x-coordinates of all local extrema in this equations please anyone could answer it for me?

f(x)=x^3+4x^2+2x

f(x)=x^4-3x^2+2x

f(x)=x^5-2x^2-4x

f(x)=x^5+4x^2-4x
 
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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