Answer:
The values are evaluated below.
Step-by-step explanation:
Given function is
[tex]-x^4-7x^3-12x^2[/tex]
We have to find the domain, range, rel max, rel min, end behaviour, increasing or decreasing intervals and zeros of polynomial.
Domain:
The domain of a function is the set of all possible values of x for the given function.
Here domain is set of all real numbers R.
Range:
The range is the resulting y-values we get after substituting all the possible x-values.From the graph we see that
The range is [tex]-\infty<y<3.124[/tex]
Relative maxima and minima of a function, are the largest and smallest value of the function on an entire domain of function.
Relative max of [tex]-x^4-7x^3-12x^2[/tex] is 0 at x=0
and [tex]\frac{3}{512}(133\sqrt57-471)[/tex] at [tex]x=\frac{-21}{8}-\frac{\sqrt57}{8}[/tex]
Relative min is
[tex]\frac{-3}{512}(471+133\sqrt57)[/tex] at [tex]x=\frac{-21}{8}+\frac{\sqrt57}{8}[/tex]
From the graph we see that
End behaviour is As [tex]\\x->\infty, f(x)->-\infty\\x->-\infty, f(x)->-\infty\\[/tex]
Increasing intervals and decreasing intervals are
Increasing: [tex]\frac{1}{8}(\sqrt57-21)<x<\frac{1}{8}(-21-\sqrt57)[/tex]
Decreasing: [tex]\frac{1}{8}(-21-\sqrt57)<x<\frac{1}{8}(\sqrt57-21)[/tex]
Zeroes are :
[tex]-x^4-7x^3-12x^2=0[/tex]
⇒ [tex](-x^2)(x^2+7x+12)=0[/tex]
⇒[tex](-x^2)(x+3)(x+4)=0[/tex]
Hence, zeroes are 0, 0, -3, -4
