Problem+32+Ari+and+Mimi

A) The function is increasing when it's derivative is positive. The derivative of f(x) is positive during the intervals of (1,6) and (8,9). Therefore the function is increasing during the intervals of (1,6) and (8,9). The function is decreasing when the slope or derivative is negative. The derivative is negative during the intervals of (0,1) and (6,8). Therefore, the function is decreasing during the intervals of (0,1) and (6,8).

B) When the first derivative goes from positive to negative the function goes from increasing to decreasing and when the function goes from increasing to decreasing there is a local maximum. The first derivative goes from positive to negative at x=6. Therefore, there is a local max at x=6. When the first derivative goes from negative to positive the function goes from decreasing to increasing and when the function goes from decreasing to increasing there is a local minimum. The first derivative goes from negative to positive at x=1 and x=8. Therefore, there is a local min at x= 1 and 8.

Ms. Sweeney: these explanations are PERFECT!!!!

C) f(x) is concave upward when the second derivative is positive. The second derivative is positive on the intervals (0, 2) and (3, 5) and (7, 9) Therefore, the f(x) is concave upward on those intervals. f(x) is concave downward when the second derivative is negative. The second derivative of f(x) is negative on the intervals (2, 3) and (5, 7). Therefore the function f(x) is concave downward on the intervals (2, 3) and (5, 7).

Ms. Sweeney: how do you know about concavity if you don't know what the second derivative is?? You don't even have the graph of the function to look at!! You need more explanation here! (and for d as well!)

D) The points of inflection are where the function changes concavity, or when the second derivative changes signs. In this case, the points of inflection occur at x=2(where the second derivative goes from positive to negative) x=3 (where the second derivative goes from negative to positive), x=5 (where the second derivative goes from positive to negative) and x=7 (where the second derivative goes from negative to positive).

E)