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f(x) is concave up whenever f'(x) is increasing. f(x) is concave down whenever f'(x) is decreasing. The test for concavity: If f"(x) is positive for all x in (a,b) then f(x) is concave up in (a,b). If f"(x) is negative for all x in (a,b) then f(x) is concave down in (a,b). A point of inflection occurs where the concavity changes. If (c, f(c)) is a point of inflection, then both #1 and #2 are true:
- f"(c) is either zero or undefined.
- f"(x) changes signs at x = c. If f"(c) (^) = 0, it doesn't guarantee that f(x) has a POI at x (^) = c.
- f(x) = x 4
- 4x 3 Find the points of inflection, then determine the intervals where the function is concave up and concave down:
- y (^) = ½x 4
- A function f is continuous on the closed interval [-1,3] and its derivatives have the values indicated in the table below. (a) Find the x-coordinates of all local extrema of f on (-1,3) and indicate which are maxima and which are minima. Explain. (b) Find all the intervals on which f is concave up or concave down. (c) Find the x-coordinates of any points of inflection. Explain. (d) Sketch the graph of y = f(x) with all the given features.
The 2nd Derivative Test In order to use this test, f'(c) must equal 0. If f"(c) > 0, then f(c) is a MINIMUM. If f"(c) (^) < 0, then f(c) is a MAXIMUM. If f"(c) = 0, then the TEST FAILS. (You'll need to go back and use the 1st Derivative Test.) Find all relative extrema. Use the 2nd Derivative Test whenever possible:
- f(x) = (-1/3) x 3
- A function is defined by f(x) = x 3
- ax 2
- bx + c, where a, b, and c are constants. f(-1) = 4 is a local maximum, and the graph has a point of inflection at x = 2. Find the value of a, b, and c:
- If g(-1) = 3, g'(-1) (^) = 0, and g"(-1) (^) = 5, what can you conclude about the value 3?
- Let h be a differentiable function with only one critical number: x = -2. If h(-2) = -4, h'(-3) = -7, and h'(-1) = 0.2, what can you conclude about the value -4?
- Based on the graph, find the sign (+,-,0) of each value: f(x) f(-2) _____ f'(-2) _____ f"(-2) _____ f(5) _____ f'(5) _____ f"(5) _____ f(0) _____ f'(0) _____ f"(0) _____
- Sketch a continuous curve with the following properties: f(-4) = 1, f(0) = 3, f(4) = 5, f'(-4) = f'(4)^ = 0, f'(x) < 0 for lxl > 4, f'(x) > 0 for lxl < 4, f"(x) > 0 for x < 0, f"(x) < 0 for x > 0.
