What are the critical values on the graph of y = x^3 - 6x^2 + 9x + 3?

Oh, critical values? Sounds like they're pretty serious business. But don't worry, I'll do my best to make them a little less critical and a little more comical for you!

So, to find the critical values of the graph y = x^3 - 6x^2 + 9x + 3, we need to find the points where the graph changes direction. In other words, where it goes from being happy to being sad, or vice versa. Just like a clown's mood at a kids' party! šŸ¤”

To find these critical values, we'll take the derivative of the equation. But hey, let's not get too derivative with the jokes here, okay?

So, taking the derivative of y = x^3 - 6x^2 + 9x + 3, we get:

dy/dx = 3x^2 - 12x + 9

Now let's set this derivative equal to zero to find the critical points, where the graph changes direction. At these points, the graph might throw a funny curveball or get a little wobbly, just like a clown on stilts! šŸ¤¹ā€ā™‚ļø

0 = 3x^2 - 12x + 9

Now, let's solve this equation to find the values of x that make the derivative equal to zero. And remember, solving equations can be a bit like juggling, so stay focused!

Dividing both sides of the equation by 3, we get:

0 = x^2 - 4x + 3

Factoring the quadratic equation, we have:

0 = (x - 3)(x - 1)

So, the critical values are x = 3 and x = 1. These are the points where the graph changes direction, just like a clown suddenly changing his funny face! šŸ¤”

And there you have it, my friend! The critical values on the graph of y = x^3 - 6x^2 + 9x + 3 are x = 3 and x = 1. Now, go and enjoy the rest of your math journey with a smile on your face, just like a clown on a unicycle! šŸŽŖ

To find the critical values of a function, we need to find the points where the derivative is equal to zero or undefined. In this case, we have the function y = x^3 - 6x^2 + 9x + 3.

Step 1: Find the derivative of the function.
The derivative of y = x^3 - 6x^2 + 9x + 3 is obtained by taking the derivative of each term:
dy/dx = 3x^2 - 12x + 9

Step 2: Set the derivative equal to zero and solve for x.
Setting 3x^2 - 12x + 9 equal to zero, we have:
3x^2 - 12x + 9 = 0

Step 3: Factor the quadratic equation.
Dividing each term by 3 gives:
x^2 - 4x + 3 = 0

Factoring the equation, we have:
(x - 1)(x - 3) = 0

Setting each factor equal to zero, we have:
x - 1 = 0 -> x = 1
x - 3 = 0 -> x = 3

Step 4: Check if the derivative is undefined at any point.
Since the derivative is a polynomial, it is defined for all real numbers. Therefore, there are no points where the derivative is undefined.

Step 5: Identify the critical values.
The critical values of the function occur at x = 1 and x = 3.

Therefore, the critical values on the graph of y = x^3 - 6x^2 + 9x + 3 are x = 1 and x = 3.

To find the critical values on the graph of the function y = x^3 - 6x^2 + 9x + 3, we first need to determine where the derivative of the function is equal to zero. The critical values correspond to the x-values where the slope of the function is zero or undefined.

To find the derivative of the function, y' with respect to x, we can differentiate each term of the function individually using the power rule. The derivative of x^3 with respect to x is 3x^2, the derivative of -6x^2 is -12x, and the derivative of 9x is 9. The constant term 3 does not contribute to the derivative since the derivative of a constant is zero.

Summing up the derivatives of each term, we get:

y' = 3x^2 - 12x + 9

Next, we set the derivative equal to zero:

3x^2 - 12x + 9 = 0

To solve this quadratic equation, we can factor it or use the quadratic formula. Let's use the quadratic formula:

x = (-b Ā± āˆš(b^2 - 4ac))/(2a)

In the equation 3x^2 - 12x + 9 = 0, a = 3, b = -12, and c = 9. Plugging in these values, we have:

x = (-(-12) Ā± āˆš((-12)^2 - 4(3)(9)))/(2(3))

Simplifying further:

x = (12 Ā± āˆš(144 - 108))/(6)
x = (12 Ā± āˆš(36))/(6)
x = (12 Ā± 6)/(6)

Simplifying the results, we have:

x1 = 3
x2 = 1

So the critical values on the graph of y = x^3 - 6x^2 + 9x + 3 are x = 3 and x = 1. These are the x-values where the slope of the function is zero or undefined, indicating potential turning points or points of inflection on the graph.

they are where

y' = 3x^2-12x+9 = 0