Question

5.1 - 2 (2 points) Let f(x) Find the open intervals on which f is concave up (down). Then determine the e-coordinates of all inflection points of 2 + 7 1. f is concave up on the intervals 2. f is concave down on the intervals 3. The inflection points occur at 2 - Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-int, 2), (3,4), or the word "none". In the last one, your answer should be a comma separated list of values or the word "none".

Answer 1

Answer is given as.

5x-2 f (x) = soll : Let xt7 0 е have which to find the open in tesrals on f is con cave up as well as down. and then a con ordinate g all inflection points of f. Now f(x) = 5x-2 at7 so flca) Call (x+7).(5) -(57-2) (1) (x+7)2 2 5x+35–5x+2 (x+72 f'(x). 37 (x+72

New, f'(x) 37 od dai 37 X - 2 (x+7)3 -74 (x+7)3 hence f"(x) 74 (x+7)3 Since , we Know that 13 function in terrals where concave up tes the f"(x) >o hence from 0 To make f "(21)70 9 ( we have to take (a + 7) < x < - 7 hence f(x) sx-2 is con cave up x +7 in the interval (-oo, 7)

and function 18 Con cave down where f"(x) Lo for the interval to make hence of" (2) Lo from 0 have to take / с е (x+7) 7o 깃 > – 7 hence the interval where fix) = su-2 x+7 is con love down is (-7,00) Since f (x) exists for all e when x+7 to hence fix) has no inflection point Candidates when a +7 to. hence f(x) = 5x-2 at7 has no inflection

point ouw x+7 *o hen a .function 5a-2 f(x)= 9 atz 0 f is con cove up the interval (-∞, -7). 2) f is Con love down the interval (-7,0). 3) has 나 no inflection points