y = 3x^5 - 10x^3 + 45x\nenter dne if an answer does not exist.\n• domain\n\n• intercepts\n o x…

y = 3x^5 - 10x^3 + 45x\nenter dne if an answer does not exist.\n• domain\n\n• intercepts\n o x - intercept(s)\n separate multiple points with a comma, e.g., (-9,0),(4,0).\n\n o y - intercept\n\n• symmetry\n about the origin about the y - axis no symmetry\n\n• asymptotes\n o horizontal asymptote\n\n o vertical asymptote\n\n• intervals of increase and decrease\n o interval of increase\n\n o interval of decease\n\n• local max and/or min\n o list all maximum points\n\n o list all minimum points\n\n• concavity and inflection points\n o interval of concave up\n\n o interval of concave down\n\n o inflections point(s)\n separate points with a comma.\n\n• graph\n using the information above, sketch the

y = 3x^5 - 10x^3 + 45x\nenter dne if an answer does not exist.\n• domain\n\n• intercepts\n o x - intercept(s)\n separate multiple points with a comma, e.g., (-9,0),(4,0).\n\n o y - intercept\n\n• symmetry\n about the origin about the y - axis no symmetry\n\n• asymptotes\n o horizontal asymptote\n\n o vertical asymptote\n\n• intervals of increase and decrease\n o interval of increase\n\n o interval of decease\n\n• local max and/or min\n o list all maximum points\n\n o list all minimum points\n\n• concavity and inflection points\n o interval of concave up\n\n o interval of concave down\n\n o inflections point(s)\n separate points with a comma.\n\n• graph\n using the information above, sketch the

Answer

Explanation:

Step1: Find the domain

A polynomial function like $y = 3x^{5}-10x^{3}+45x$ is defined for all real - numbers. $(-\infty,\infty)$

Step2: Find the x - intercepts

Set $y = 0$, so $3x^{5}-10x^{3}+45x=0$. Factor out $x$: $x(3x^{4}-10x^{2}+45)=0$. Let $u = x^{2}$, then $3u^{2}-10u + 45=0$. The discriminant of $3u^{2}-10u + 45$ is $\Delta=(-10)^{2}-4\times3\times45=100 - 540=-440<0$. The only real root of $3x^{5}-10x^{3}+45x = 0$ is $x = 0$. So the x - intercept is $(0,0)$.

Step3: Find the y - intercept

Set $x = 0$ in $y = 3x^{5}-10x^{3}+45x$. Then $y=0$.

Step4: Check for symmetry

Replace $x$ with $-x$: $y = 3(-x)^{5}-10(-x)^{3}+45(-x)=-3x^{5}+10x^{3}-45x=-(3x^{5}-10x^{3}+45x)$. The function is odd, so it is symmetric about the origin.

Step5: Find the asymptotes

Since it is a polynomial function, the degree is 5 (an odd degree polynomial), there are no horizontal or vertical asymptotes. $DNE$ for both horizontal and vertical asymptotes.

Step6: Find the intervals of increase and decrease

First, find the derivative $y'=15x^{4}-30x^{2}+45$. Let $t = x^{2}$, then $y'=15t^{2}-30t + 45=15(t^{2}-2t + 3)$. The discriminant of $t^{2}-2t + 3$ is $\Delta=(-2)^{2}-4\times3=-8<0$, and the coefficient of $t^{2}$ is positive. So $y'>0$ for all $x\in(-\infty,\infty)$. The interval of increase is $(-\infty,\infty)$ and the interval of decrease is $DNE$.

Step7: Find local maxima and minima

Since the function is always increasing, there are no local maxima or minima. $DNE$ for both maximum and minimum points.

Step8: Find concavity and inflection points

Find the second - derivative $y'' = 60x^{3}-60x=60x(x^{2}-1)=60x(x - 1)(x + 1)$. Set $y''>0$: $60x(x - 1)(x + 1)>0$. The solution is $(-1,0)\cup(1,\infty)$ (interval of concave up). Set $y''<0$: $60x(x - 1)(x + 1)<0$. The solution is $(-\infty,-1)\cup(0,1)$ (interval of concave down). Set $y'' = 0$, then $x=-1,0,1$. The inflection points are $(-1,-38),(0,0),(1,38)$.

Answer:

  • Domain: $(-\infty,\infty)$
  • x - intercept(s): $(0,0)$
  • y - intercept: $0$
  • Symmetry: About the origin
  • Horizontal Asymptote: $DNE$
  • Vertical Asymptote: $DNE$
  • Interval of Increase: $(-\infty,\infty)$
  • Interval of Decrease: $DNE$
  • List all maximum points: $DNE$
  • List all minimum points: $DNE$
  • Interval of Concave up: $(-1,0)\cup(1,\infty)$
  • Interval of Concave Down: $(-\infty,-1)\cup(0,1)$
  • Inflections Point(s): $(-1,-38),(0,0),(1,38)$