Problem+15+Genevieve+and+Sylvia



More on part 15c:

If there were inflection points, one would find them by deriving the second derivative, then setting that to zero. f "(x) = 4e^2x + e^-x = 0 4e^2x = -(1/e^x) 4e^3x = -1 e^3x = -1/4 3x lne = ln (-1/4) However, it is impossible to take the natural log of a negative number, so it is impossible to find an inflection point where the graph switched from concave up to concave down or the other way around. So to find whether the entire function is concave up or down, plug any point into the second derivative 4e^(2*2) + e^-2 = 218.528 This number (or any number solved through this equation) is positive. Since the second derivative is positive, the function is concave up.  Ms. Sweeney: My only comment is that I got a different y-value whe I plugged (-1/3)ln2 into the function. e^(2)(-1/3ln2) is the same as e^[(ln2)^(-2/3), which is equal to 2^(-2/3), which is 4^(-1/3)...Otherwise, very good, and convincing and clear and succinct argument!