College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Find the transition points for the function [tex]y = 3x^5 - 45x^3[/tex].

1. The critical point(s) at [tex]x = \square[/tex]

2. The point(s) of inflection at [tex]x = \square[/tex]

(Use symbolic notation and fractions where needed. Provide your answers in the form of comma-separated values.)

Answer :

Sure! Let's find the critical points and points of inflection for the function [tex]\( y = 3x^5 - 45x^3 \)[/tex].

### Critical Points

1. Find the First Derivative: The first step is to find the derivative of the function with respect to [tex]\( x \)[/tex]:
[tex]\[
y' = \frac{d}{dx}(3x^5 - 45x^3) = 15x^4 - 135x^2
\][/tex]

2. Set the First Derivative to Zero: Solve the equation [tex]\( y' = 0 \)[/tex] to find the critical points:
[tex]\[
15x^4 - 135x^2 = 0
\][/tex]

3. Factor the Equation: Let's factor the equation:
[tex]\[
15x^2(x^2 - 9) = 0
\][/tex]
This gives us two solutions:
[tex]\[
x^2 = 0 \quad \text{or} \quad x^2 - 9 = 0
\][/tex]
Solving these:
[tex]\[
x = 0 \quad \text{or} \quad x = \pm 3
\][/tex]

4. Critical Points: Therefore, the critical points are at:
[tex]\[
x = -3, 0, 3
\][/tex]

### Points of Inflection

1. Find the Second Derivative: Next, take the derivative of [tex]\( y' \)[/tex] to find the second derivative:
[tex]\[
y'' = \frac{d}{dx}(15x^4 - 135x^2) = 60x^3 - 270x
\][/tex]

2. Set the Second Derivative to Zero: Solve the equation [tex]\( y'' = 0 \)[/tex] to find the points of inflection:
[tex]\[
60x^3 - 270x = 0
\][/tex]

3. Factor the Equation: Factor the second derivative:
[tex]\[
60x(x^2 - \frac{27}{6}) = 0
\][/tex]
Solving this:
[tex]\[
x = 0 \quad \text{or} \quad x = \pm \frac{3\sqrt{2}}{2}
\][/tex]

4. Points of Inflection: The points of inflection are at:
[tex]\[
x = 0, -\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}
\][/tex]

### Final Answer
- The critical points are at [tex]\( x = -3, 0, 3 \)[/tex].
- The points of inflection are at [tex]\( x = 0, -\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2} \)[/tex].