Question

determine the following characteristics of the graph.
y = 3x⁵ - 10x³ + 45x
type dne if the answer does not exist for any of the characteristics below

• domain
• intercepts
• x - intercept(s)

if the are multiple points, separate with a comma. ex. (-9,0),(4,0)

• y - intercept
• symmetry

select an answer

• asymptotes
• horizontal asymptote
• vertical asymptote
• intervals of increase and decrease
• interval of increase
• interval of decease
• local max and/or min
• list all maximum points
• list all minimum points

Explanation:

Step1: Find the Domain

The function \( y = 3x^5 - 10x^3 + 45x \) is a polynomial function. Polynomial functions are defined for all real numbers. So the domain is all real numbers, which can be written as \( (-\infty, \infty) \).

Step2: Find the x - intercepts

To find the x - intercepts, we set \( y = 0 \), so we solve the equation \( 3x^5 - 10x^3 + 45x=0 \). Factor out an \( x \): \( x(3x^4 - 10x^2 + 45)=0 \). The quadratic in terms of \( u = x^2 \) is \( 3u^2-10u + 45 = 0 \). The discriminant of this quadratic is \( \Delta=(- 10)^2-4\times3\times45=100 - 540=- 440<0 \). So the only real solution for \( 3x^4 - 10x^2 + 45 = 0 \) is none, and the solution for \( x = 0 \) is \( x = 0 \). So the x - intercept is \( (0,0) \).

Step3: Find the y - intercept

To find the y - intercept, we set \( x = 0 \) in the function. Substitute \( x = 0 \) into \( y=3x^5 - 10x^3 + 45x \), we get \( y=3(0)^5-10(0)^3 + 45(0)=0 \). So the y - intercept is \( (0,0) \).

Step4: Check for Symmetry

A function \( f(x) \) is odd if \( f(-x)=-f(x) \) and even if \( f(-x)=f(x) \). Let's find \( f(-x) \): \( f(-x)=3(-x)^5-10(-x)^3 + 45(-x)=-3x^5 + 10x^3-45x=-(3x^5 - 10x^3 + 45x)=-f(x) \). So the function is odd, and it is symmetric about the origin.

Step5: Find Asymptotes

For polynomial functions of degree \( n\geq1 \), there are no horizontal or vertical asymptotes. Because as \( x
ightarrow\pm\infty \), the leading term \( 3x^5 \) dominates, and the function does not approach a constant (so no horizontal asymptote) and the function is defined for all real numbers (so no vertical asymptote). So horizontal asymptote: DNE, vertical asymptote: DNE.

Step6: Find Intervals of Increase and Decrease

First, find the derivative of the function. The derivative \( y^\prime=f^\prime(x)=15x^4-30x^2 + 45 \). Factor out 15: \( y^\prime=15(x^4 - 2x^2 + 3) \). Let \( u = x^2 \), then the quadratic in \( u \) is \( u^2-2u + 3 \). The discriminant of this quadratic is \( \Delta=(-2)^2 - 4\times1\times3=4 - 12=-8<0 \). And the coefficient of \( u^2 \) is positive, so \( x^4 - 2x^2 + 3>0 \) for all real \( x \). Then \( y^\prime=15(x^4 - 2x^2 + 3)>0 \) for all real \( x \). So the function is increasing on \( (-\infty,\infty) \) and there is no interval of decrease.

Step7: Find Local Max and Min

Since the derivative \( y^\prime>0 \) for all real \( x \), the function is always increasing. So there are no local maximum or minimum points.

Answer:

s:

• Domain: \( (-\infty, \infty) \)
• x - intercept(s): \( (0,0) \)
• y - intercept: \( (0,0) \)
• Symmetry: Symmetric 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