Answer
A. The domain is $(-\infty, \infty)$
B. The y-intercept is $0$
The x-intercepts are $0$ and $\pm \sqrt[4] 5$
C. The function is an odd function.
D. $\lim\limits_{x \to -\infty} (x^5-5x) = \lim\limits_{x \to -\infty} (x)(x^4-5)=-\infty$
$\lim\limits_{x \to \infty} (x^4-4x) = \lim\limits_{x \to \infty} (x)(x^4-5)=\infty$
There are no asymptotes.
E. The function is increasing on the intervals $(-\infty, -1)\cup (1, \infty)$
The function is decreasing on the interval $(-1,1)$
F. $(1,-4)$ is a local minimum.
$(-1,4)$ is a local maximum.
G. The graph is concave down on the interval $(-\infty,0)$
The graph is concave up on the interval $(0, \infty)$
The point of inflection is $(0,0)$
H. We can see a sketch of the curve below.
Work Step by Step
$y = x^5-5x$
A. The function is defined for all real numbers. The domain is $(-\infty, \infty)$
B. When $x = 0,$ then $y = (0)^5-5(0) = 0$
The y-intercept is $0$
When $y = 0$:
$x^5+5x = 0$
$x(x^4-5) = 0$
$x = 0$ or $x = \pm \sqrt[4] 5$
The x-intercepts are $0$ and $\pm \sqrt[4] 5$
C. $(-x)^5-5(-x) = -(x^5-5x)$
The function is an odd function.
D. $\lim\limits_{x \to -\infty} (x^5-5x) = \lim\limits_{x \to -\infty} (x)(x^4-5)=-\infty$
$\lim\limits_{x \to \infty} (x^4-4x) = \lim\limits_{x \to \infty} (x)(x^4-5)=\infty$
There are no asymptotes.
E. We can find the values of $x$ such that $y' = 0$:
$y' = 5x^4-5 = 0$
$x^4-1 = 0$
$(x^2-1)(x^2+1) = 0$
$x^2-1 = 0$ or $x^2+1 = 0$
$x = -1, 1$
The function is increasing on the intervals $(-\infty, -1)\cup (1, \infty)$
The function is decreasing on the interval $(-1,1)$
F. When $x = 1$, then $y = (1)^5-5(1)= -4$
$(1,-4)$ is a local minimum.
When $x = -1$, then $y = (-1)^5-5(-1)= 4$
$(-1,4)$ is a local maximum.
G. We can find the values of $x$ such that $y'' = 0$:
$y'' = 20x^3 = 0$
$x = 0$
When $x \lt 0$, then $y'' \lt 0$
The graph is concave down on the interval $(-\infty,0)$
When $x \gt 0$, then $y'' \gt 0$
The graph is concave up on the interval $(0, \infty)$
When $x = 0,$ then $y = (0)^5-5(0) = 0$
The point of inflection is $(0,0)$
H. We can see a sketch of the curve below.