The graph of y = 3x^2 - x^3 has a relative maximum at what point?
To find the relative maximum of a function, we need to find the critical points where the derivative of the function equals zero. Let's find the derivative of the function:
y = 3x^2 - x^3
To find the derivative, we use the power rule:
dy/dx = d/dx(3x^2 - x^3) = 6x - 3x^2
Setting the derivative equal to zero:
6x - 3x^2 = 0
Factoring out an x:
x(6 - 3x) = 0
To find the critical points, we solve for x:
x = 0 (from x = 0)
or
6 - 3x = 0 (from 6 - 3x = 0)
Solving the equation 6 - 3x = 0:
6 - 3x = 0
-3x = -6
x = 2
So, the critical points of the function are x = 0 and x = 2. To determine if these are relative maximum or minimum points, we can analyze the second derivative of the function.
To find the relative maximum of a function, we need to analyze the critical points and determine their nature using the second derivative test.
Let's start by finding the critical points of the function y = 3x^2 - x^3. Critical points occur where the derivative of the function is zero or undefined.
First, we'll take the derivative of the function:
dy/dx = d/dx (3x^2 - x^3)
= 6x - 3x^2
Setting the derivative equal to zero, we have:
6x - 3x^2 = 0
Factoring out x, we get:
x(6 - 3x) = 0
So, we have two possibilities:
1) x = 0
2) 6 - 3x = 0
Solving the second equation, we find:
6 - 3x = 0
3x = 6
x = 2
So, we have two critical points: x = 0 and x = 2.
Next, we need to determine the nature of these critical points using the second derivative test.
To do this, we find the second derivative by differentiating the previous derivative:
d^2y/dx^2 = d/dx(6x - 3x^2)
= 6 - 6x
Now, we substitute the critical points into the second derivative:
For x = 0:
d^2y/dx^2 = 6 - 6(0)
= 6
For x = 2:
d^2y/dx^2 = 6 - 6(2)
= 6 - 12
= -6
According to the second derivative test:
- If the second derivative is positive at a critical point, the function has a relative minimum at that point.
- If the second derivative is negative at a critical point, the function has a relative maximum at that point.
From our calculations:
- At x = 0, the second derivative is positive (6), which means the function has a relative minimum at x = 0.
- At x = 2, the second derivative is negative (-6), which means the function has a relative maximum at x = 2.
Therefore, the graph of y = 3x^2 - x^3 has a relative maximum at the point (2, f(2)), or (2, 8).