Question

find the relative maximas and minimas of each polynomial. (f(x)=-x^{4}+3x^{2}-2x + 1). round each answer to the nearest tenth! please do not put any spaces in between the numbers and the comma. please put the maxima with the smallest x - value first. minima: maxima: maxima:

Explanation:

Step1: Find the derivative of the function

Given \(f(x)=-x^{4}+3x^{2}-2x + 1\), using the power - rule \((x^n)^\prime=nx^{n - 1}\), we have \(f^\prime(x)=-4x^{3}+6x - 2\).

Step2: Set the derivative equal to zero

\(-4x^{3}+6x - 2 = 0\). Factor out \(-2\) to get \(2x^{3}-3x + 1=0\). By trial - and - error, we find that \(x = 1\) is a root. Then we can factor \(2x^{3}-3x + 1=(x - 1)(2x^{2}+2x - 1)\).
Using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(2x^{2}+2x - 1\) where \(a = 2\), \(b = 2\), \(c=-1\), we get \(x=\frac{-2\pm\sqrt{4+8}}{4}=\frac{-2\pm2\sqrt{3}}{4}=\frac{-1\pm\sqrt{3}}{2}\). So the critical points are \(x = 1\), \(x=\frac{-1+\sqrt{3}}{2}\approx0.37\), \(x=\frac{-1 - \sqrt{3}}{2}\approx - 1.37\).

Step3: Use the second - derivative test

Find the second - derivative \(f^{\prime\prime}(x)=-12x^{2}+6\).
For \(x=\frac{-1+\sqrt{3}}{2}\), \(f^{\prime\prime}(\frac{-1+\sqrt{3}}{2})=-12(\frac{-1 + \sqrt{3}}{2})^{2}+6=-12(\frac{1-2\sqrt{3}+3}{4})+6=-3(4 - 2\sqrt{3})+6=-12 + 6\sqrt{3}+6=6\sqrt{3}-6\approx4.39>0\), so \(f(x)\) has a local minimum at \(x=\frac{-1+\sqrt{3}}{2}\), and \(f(\frac{-1+\sqrt{3}}{2})=-(\frac{-1+\sqrt{3}}{2})^{4}+3(\frac{-1+\sqrt{3}}{2})^{2}-2(\frac{-1+\sqrt{3}}{2})+1\approx0.2\).
For \(x=\frac{-1 - \sqrt{3}}{2}\), \(f^{\prime\prime}(\frac{-1 - \sqrt{3}}{2})=-12(\frac{-1 - \sqrt{3}}{2})^{2}+6=-12(\frac{1 + 2\sqrt{3}+3}{4})+6=-3(4 + 2\sqrt{3})+6=-12-6\sqrt{3}+6=-6 - 6\sqrt{3}\approx - 16.39<0\), so \(f(x)\) has a local maximum at \(x=\frac{-1 - \sqrt{3}}{2}\), and \(f(\frac{-1 - \sqrt{3}}{2})=-(\frac{-1 - \sqrt{3}}{2})^{4}+3(\frac{-1 - \sqrt{3}}{2})^{2}-2(\frac{-1 - \sqrt{3}}{2})+1\approx4.8\).
For \(x = 1\), \(f^{\prime\prime}(1)=-12\times1^{2}+6=-6<0\), so \(f(x)\) has a local maximum at \(x = 1\), and \(f(1)=-1^{4}+3\times1^{2}-2\times1 + 1=1\).

Answer:

Maxima: \(-1.4,4.8\); Maxima: \(1,1\); Minima: \(0.4,0.2\)