Question

3. A mor norrean fco flo fío f' f'>0. typical function, derivative of functia alternates For zeros of a sign of the slope of the the te B). Use the linear approximation to show that it fal=0 & f'lad>o the for xxa but very close to a - fx) < o exca, x-a small & flx) >0 (xad, x-a small C). Conclude that it is impossible for a function f(x) with nice derivatives to have two cobsecutive zeros x, < x2 for which f'(x) > 0 8 f'(x2170. "Consecutive zeros: f(x)=0, f(x2150 f(x) to it xcx <xz.

Answer 1

SY (B) Using linear approximation of a funetion f(e) around a point a one can write fbe) ü f(a) + f'(a) (-a). - 0 Now, given to f (a) = 0, f'(a) >0 . If x<a (x-a small) i.e. (al-a) <0, then from 1 one can see f(x) <0. If xxa (x-a small) i.e. (x-a) 3o, then from it can be said that f(x) >0. (c) set f(a) be a function having nice derivatives with two consecutive Zeroes X7 and xy If f'() >0 then foe) is increasing around the point x=ds. Now, since there is no zero between x, and He of flow f (ve) must be decreasing around the point o x=ng so, f(x) <0. Hence, it can be concluded that there is no such f(x) for which f(x) >0 and of' (2) >0 for two consecutive zeroes xe, da