Question

(1 point) Given the function f:1-5, 1] + [1,1]; f(x) = cos x, check which one(s) of the properties it has. A. strictly decreasing B. decreasing C. injective D. surjective E. strictly increasing F. increasing G. None of the above

Answer 1

Solo 15 ] delfined by Fix) = 605% Y-axis -IT- O X-aris @ Strictly decreasing : Fox) = cosx flix) = -simx < (O, T) > 16,0) .fra z lose is increasing in (1,0) and decreasing (0,1). Thus f(x) = tosx is not stoictly increasing on [1,1] ® Decreasing a I'(X) = - Sinx < (0) > [-1,0) s x) Pis Pnereasong on [ 10) and decreasing in costlyu) fo) = cosx Ps not decreasing on [1,1].

© Injective : Fan graph () = f() But 1 4 + - 1 1 fru= cosx is not injettive on T-4,51: © Surjective: Intermediate Thus fex) attains bounds and I There fode by value theorem (If a continous function attains its bounds then it will take every value between bounds ).. & f(x) = cost attains every value between 1 and li fox) = cosx ls Surjective. © Strictly increasing 1o) = Cose → f'(x) = -Smix <0 (0,) so (-5,0) • Fox) = cose is increasing im (0,) and decreasing on 14,0) fo) is not strictly increasing:

Pncreasing F(x) = Cosn - I'(x) = - Sinx Lo (o, I) fixe) is increasing im (01) and decreasing on L o) fox) is not increasing. Thank you