Find the points on the graph of f where the tangent line is horizontal.
f(x) = x3 - 15x2
(x, y) = ( , ) (smaller x-value)
(x, y) = ( , ) (larger x-value)
To find the points on the graph of f where the tangent line is horizontal, we need to find the values of x for which the derivative of f is equal to zero.
Step 1: Find the derivative of f(x).
f'(x) = 3x^2 - 30x
Step 2: Set f'(x) = 0 and solve for x.
3x^2 - 30x = 0
3x(x - 10) = 0
From this equation, we can see that either 3x = 0 or (x - 10) = 0.
- If 3x = 0, then x = 0.
- If (x - 10) = 0, then x = 10.
So the possible x-values where the tangent line is horizontal are x = 0 and x = 10.
Step 3: Plug these x-values into f(x) to find the corresponding y-values.
For x = 0, we calculate f(0):
f(0) = (0)^3 - 15(0)^2
f(0) = 0 - 0
f(0) = 0
So the point (0, 0) is on the graph of f.
For x = 10, we calculate f(10):
f(10) = (10)^3 - 15(10)^2
f(10) = 1000 - 1500
f(10) = -500
So the point (10, -500) is on the graph of f.
Therefore, the points on the graph of f where the tangent line is horizontal are:
(0, 0) and (10, -500).