a) Describe the graph of ƒ(x) using the terminology: increasing, decreasing, local maximum or local minimum points, concavity, points of inflection and the following information
ƒ′(x) < 0 when x < 3
ƒ′(x) = 0 when x = 3
ƒ′(x) > 0 when x > 3
Choose the graph that could be a graph of ƒ(x) and justify your answer.
the graph of y = x^2 - 6x + 9
the graph of y = x^2 - 6x + 9
the graph of y = x^2 - 6x + 9
b) Describe the graph of ƒ(x) using the terminology: increasing, decreasing, local maximum or local minimum points, concavity, points of inflection and the following information
ƒ′(x) < 0 when x < 2
ƒ′(x) = 0 when x = 2
ƒ′(x) > 0 when x > 2
Choose the graph that could be a graph of ƒ(x) and justify your answer.
the graph of y = x^2 + 4x + 4
the graph of y = x^2 + 4x + 4
the graph of y = x^2 + 4x + 4
c) Describe the graph of ƒ(x) using the terminology: increasing, decreasing, local maximum or local minimum points, concavity, points of inflection and the following information
ƒ′(x) < 0 when x < -1
ƒ′(x) = 0 when x = -1
ƒ′(x) > 0 when -1 < x < 1
Choose the graph that could be a graph of ƒ(x) and justify your answer.
the graph of y = -x^3 - 3x
the graph of y = -x^3 - 3x
the graph of y = -x^3 - 3x
a) The graph that could be a graph of ƒ(x) is y = x^2 - 6x + 9. This is because the derivative is negative when x is less than 3 and positive when x is greater than 3, which indicates that the function is decreasing to the left of the local maximum at x = 3 and increasing to the right of it. Additionally, the derivative is 0 at the point x = 3, where the function has a local maximum. The concavity of the function is upward, and there are no points of inflection. This matches the characteristics of the parabola y = x^2 - 6x + 9.
b) The graph that could be a graph of ƒ(x) is y = x^2 + 4x + 4. This is because the derivative is negative when x is less than 2 and positive when x is greater than 2, which indicates that the function is decreasing to the left of the local minimum at x = 2 and increasing to the right of it. Additionally, the derivative is 0 at the point x = 2, where the function has a local minimum. The concavity of the function is upward, and there are no points of inflection. This matches the characteristics of the parabola y = x^2 + 4x + 4.
c) The graph that could be a graph of ƒ(x) is y = -x^3 - 3x. This is because the derivative is negative when x is less than -1 and positive when x is between -1 and 1, which indicates that the function is decreasing to the left of the point x = -1 and increasing to the right of it until it reaches a local maximum at x = 0. The derivative is 0 at x = -1, which is a critical point where the function changes from decreasing to increasing. The concavity of the function is downward, and there is a point of inflection at x = 0 where the concavity changes from downward to upward. This matches the characteristics of the cubic function y = -x^3 - 3x.