Question

1. (20 points) The second derivative of a function f(x) satisfies f "(x) = 10x4 - 2 Moreover, f'(0) = 0 and f(1) = 0. (a) Find the function f(x). (b) Draw a graph of f(x). Indicate all asymptotes (if any), local maxima and minima, inflection points, intervals where f(x) is increasing, and intervals where f(x) is concave upward.

Answer 1

at x=0 f'(x) = 10x7 - 2 f (0)=-2 co at x =0 have a maximum. at x=1 f''(I) = 10-2 > 0 so at x = 1 have a minimum. at x = -1 f!(-1)= 10-2 70 at x=-1 have a minimum. For point of inflection peut 102e9-220 f(x) 20. so If f(x) > 0 on point of inflection then concave upward. f (x) <0 on point of inflection the concave downward so (oo, – 5t) u (5 ,0) - Concave upward. (54 54) Concave down worde

for increasing and decreasing (-1,0) U (1,00) - increasing (-00,-1) 0 (0,1) - decreasing. So Now graph of f(x) = x - x² + 2 (minimum) (maximum) -I 15 -54 th minimny (point of point of inflection inflection)