Problem+35+Marnie+and+Alex+G

 35: f(x) = 2 + 2x² - x⁴ f ‘(x) = 4x - 4x³ f ‘‘(x) = -12x²

a) When plugging in 0 for f ‘(x) we found that x= ± 1, 0, than using the x coordinates we plugged in values lower and found that below -1 the values were (+), between -1, 0 they were (-), between 0, 1 the values were (+) and than after 1 they were (-). With this information we knew that whenever it was (-) it was decreasing and when it was (+) it was increasing

Increasing over the intervals of: (-∞,-1), (0,1) Decreasing over the interval: (-1,0), (1,∞)

b) We plugged in 0 for the f ‘(x) and found x= ± 1, 0 than we knew when it was local max because the function would change from (+) to (-) at that point or increasing to decreasing or vice versa (-) to (+) or decreasing to increasing

Local Maximum at x=-1, 1 Local Minimum at x=0

c) We then found the second derivative of the function to find when those values where (-) or (+) to determine when the function was concave up or down and the inflection points. When plugging 0 into f ‘‘(x) = -12x² we found that x= 1/√3, -1/√3 so we plugged in values between those two points and found that between (-∞, -1/√3) and (1/√3,∞) the values where (-) and between (-1/√3,1/√3) they were (+) therefore when ever they were (-) they were concave down and when they were (+) they were concave up

Concave up (-1/√3,1/√3) Concave down (-∞, -1/√3), (1/√3,∞) Inflection Point (±1/√3, 23/9)

d)

-Alex Gerson and Marnie Wachs

Ms. Sweeney: you need more explanation for my liking!! What is the derivative? how did you come up with the above conclusions, based on your knowledge of calculus?