Connecting F F' F" Graphs

What will be true at an inflection point? (select the best answer)

• f(x)=0 and there is a sign change

• f'(x)=0 and there is a sign change

• f''(x)=0 and there is a sign change

• The function is increasing then decreasing

Let f(t) be the depth (cm) of the water in a tank at time t (minutes). Using correct units, explain the best meaning of the statement f '(30) = 20.

• During the first 30 minutes, the water in the tank rises 20 cm.

• After 30 minutes, the water in the tank is up to 20 cm in depth.

• At 30 minutes, the water in the tank is increasing to 20 cm.

• At 30 minutes, the water in the tank is increasing at 20 cm/min

The function shown is the graph of f '(x), the derivative of f(x). The domain is [-3,5]. On which interval is the graph of f(x) concave upward?

• (-1,1) & (3,5]

• [-3,-2) & (4,5]

• [-3,-1) & (1,3)

• (-2,1) & (1,4)

This graph is of the second derivative, f" (x). For what value of x does the first derivative, f '(x) have a local maximum?

• x = 0

• x = 1

• x = 2

• x = 4

• x = 5

From the graph on the left it can be seen that the graph f' on the interval (5,9) is negative, decreasing and concave down. Based on these observations it can be concluded that:

• f is decreasing and concave up, f" is negative

• f is increasing and concave down, f" is positive

• f is increasing and concave down, f" is positive

• f is decreasing and concave down, f" is negative

The twice-differentiable function f is shown to the left on the domain (-9,9). The function f has points of inflection at x = -7.8, x = -3.6, x = 1.2, and x = 7.6. Determine what could be said about the values of f(-5), f'(-5), f"(-5).

• f(-5) < f'(-5) < f"(-5)

• f'(-5) < f(-5) < f"(-5)

• f(-5) < f"(-5) < f'(-5)

• f"(-5) < f'(-5) < f(-5)

The twice-differentiable function f is shown to the left on the domain (-9,9). The function f has points of inflection at x = -4, x = -1, x = 2.4, and x = 8. Determine what could be said about the values of f(-1), f'(-1), f"(-1).

• f"(-1) < f(-1) < f'(-1)

• f"(-1) < f'(-1) < f(-1)

• f(-1) < f"(-1) < f'(-1)

• f'(-1) < f"(-1) < f(-1)

The graph of f', the derivative of f is shown to the left. Determine the values of x, if any, at which the function f has an inflection point on the interval (-9,9).

The graph of f', the derivative of f is shown to the left. Determine all the intervals on which the function f is concave down on the interval (-9,9).

From the graph on the left it can be seen that the function f on the interval (-2,0) is positive, increasing and concave down. Based on these observations it can be concluded that:

• f' is positive and increasing, f" is negative

• f' is positive and decreasing, f" is negative

• f' is negative and decreasing, f" is negative

• f' is positive and decreasing, f" is positive