Question

find the relative maximas and minimas of each polynomial. f(x)=x^4 + 6x^3-11x^2 + 8x - 2. 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

The function is $f(x)=x^{4}+6x^{3}-11x^{2}+8x - 2$. Using the power - rule $(x^n)^\prime=nx^{n - 1}$, the derivative $f^\prime(x)=4x^{3}+18x^{2}-22x + 8$.

Step2: Set the derivative equal to zero

$4x^{3}+18x^{2}-22x + 8 = 0$. Divide through by 2 to simplify: $2x^{3}+9x^{2}-11x + 4 = 0$. We can try to find rational roots using the Rational Root Theorem. The possible rational roots are factors of $\frac{4}{2}=2$, so $\pm1,\pm2$. By testing, we find that $x = 1$ is a root.

Step3: Factor the polynomial

Since $x = 1$ is a root, we can factor $2x^{3}+9x^{2}-11x + 4$ as $(x - 1)(2x^{2}+11x - 4)$. Then we factor $2x^{2}+11x - 4=(2x - 1)(x + 4)$. So $f^\prime(x)=(x - 1)(2x - 1)(x + 4)$.

Step4: Find the critical points

Set $f^\prime(x)=0$. The critical points are $x=-4,\frac{1}{2},1$.

Step5: Use the second - derivative test

Find the second - derivative $f^{\prime\prime}(x)=12x^{2}+36x-22$.
$f^{\prime\prime}(-4)=12\times(-4)^{2}+36\times(-4)-22=192-144 - 22 = 26>0$, so $x=-4$ is a local minimum.
$f^{\prime\prime}(\frac{1}{2})=12\times(\frac{1}{2})^{2}+36\times\frac{1}{2}-22=3 + 18-22=-1<0$, so $x=\frac{1}{2}$ is a local maximum.
$f^{\prime\prime}(1)=12\times1^{2}+36\times1-22=12 + 36-22=26>0$, so $x = 1$ is a local minimum.

Step6: Find the function values at the critical points

$f(-4)=(-4)^{4}+6\times(-4)^{3}-11\times(-4)^{2}+8\times(-4)-2=256-384-176-32 - 2=-338$.
$f(\frac{1}{2})=(\frac{1}{2})^{4}+6\times(\frac{1}{2})^{3}-11\times(\frac{1}{2})^{2}+8\times\frac{1}{2}-2=\frac{1}{16}+\frac{6}{8}-\frac{11}{4}+4 - 2=\frac{1 + 12-44 + 64 - 32}{16}=\frac{1}{16}\approx0.1$.
$f(1)=1^{4}+6\times1^{3}-11\times1^{2}+8\times1-2=1 + 6-11 + 8-2=2$.

Answer:

Maxima: 0.1
Maxima:
Minima: - 338
Minima: 2