Question

given the polynomial function, determine the symmetry and positive interval (round to the nearest hundredth.)
$f(x) = -x^4 + 9x^2 - 6$
$square$ positive: (0, -2)(2, 0)
$square$ positive: (-2.88, -0.85) (0.85, 2.88)
$square$ symmetry: odd
$square$ symmetry: even

Explanation:

Step1: Find roots of $f(x)=0$

Set $-x^4 + 9x^2 - 6 = 0$. Let $u=x^2$, so $-u^2 + 9u - 6 = 0$, or $u^2 - 9u + 6 = 0$.
Use quadratic formula: $u=\frac{9\pm\sqrt{81-24}}{2}=\frac{9\pm\sqrt{57}}{2}$
$\sqrt{57}\approx7.55$, so $u_1=\frac{9+7.55}{2}\approx8.275$, $u_2=\frac{9-7.55}{2}\approx0.725$
Solve for $x$: $x=\pm\sqrt{8.275}\approx\pm2.88$, $x=\pm\sqrt{0.725}\approx\pm0.85$

Step2: Test sign of $f(x)$

Test $x=1$: $f(1)=-1+9-6=2>0$, so positive between $\pm0.85$ and $\pm2.88$.
Test $x=0$: $f(0)=-6<0$, so negative between $0$ and $\pm0.85$.
Test $x=3$: $f(3)=-81+81-6=-6<0$, so negative outside $\pm2.88$.

Step3: Check function symmetry

Compute $f(-x)=-(-x)^4+9(-x)^2-6=-x^4+9x^2-6=f(x)$, so even function.

Answer:

Positive: $(-2.88, -0.85) \cup (0.85, 2.88)$
Symmetry: Even