Please help I want to post a picture but Jiskha doesn't allow me so I just describe what I see.the derivative tangent line curve start at (0,1/2). It curve downward pass the x-axis at (1,0)until it reach a local minimum at (2,-1)then it curve upward until it reach a local maximum at(4,1)then it curve downward pass the x-axis at (5,0) and it stop at (6,-2)
The graph of a function f is shown.
(a) On what intervals is f increasing? (Enter your answer using interval notation.)
On what intervals is f decreasing? (Enter your answer using interval notation.)
(b) At what values of x does f have a local maximum or minimum? (Enter your answers as a comma-separated list.)
x =
a) f increase at(0,1]U[3,5)
a)f decrease at (1,3]U(5,6]
b) local max and min are at 1,3,5
(a) To determine where the function f is increasing or decreasing, we need to analyze the behavior of its derivative. The derivative provides information about the slope of the tangent line to the curve at different points.
Let's go through the given information and find the critical points and intervals of increasing and decreasing.
From the given information:
- The tangent line starts at (0, 1/2).
- It curves downward and passes the x-axis at (1, 0).
- It reaches a local minimum at (2, -1).
- It curves upward and reaches a local maximum at (4, 1).
- It curves downward and passes the x-axis at (5, 0).
- It stops at (6, -2).
To find the intervals where f is increasing or decreasing, we need to locate the critical points (where the derivative is zero or undefined) and analyze the sign of the derivative in the intervals between these points.
Let's analyze the behavior of f between these critical points:
- Between x = 0 and x = 1: The function is decreasing since the curve is curving downward.
- Between x = 1 and x = 2: The function is increasing since the curve starts curving upward.
- Between x = 2 and x = 4: The function is decreasing since it reaches a local minimum at (2, -1).
- Between x = 4 and x = 5: The function is increasing since it reaches a local maximum at (4, 1).
- Between x = 5 and x = 6: The function is decreasing since the curve is curving downward.
Therefore, we can conclude:
- f is increasing on the interval (1, 2) and (4, 5) (excluding the endpoints).
- f is decreasing on the interval (0, 1), (2, 4), (5, 6) (excluding the endpoints).
(b) To find the values of x where f has a local maximum or minimum, we need to look for the critical points, where the derivative is zero or undefined.
From the given information:
- The function reaches a local minimum at (2, -1).
- The function reaches a local maximum at (4, 1).
Therefore, the values of x where f has a local maximum or minimum are x = 2, and x = 4.