Solução_Calculo_Stewart_6e

F.
TX.10 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH ¤ 157
13. (a) f(x) =sinx +cosx, 0 ≤ x ≤ 2π. f 0 (x) =cosx − sin x =0 ⇒ cos x =sinx ⇒ 1= sin x
cos x
tan x =1 ⇒ x = π 4 or 5π 4 . Thus, f 0 (x) > 0 ⇔ cos x − sin x>0 ⇔ cos x>sin x ⇔ 0 ln 1 2
⇔
x> 1 (ln 1 − ln 2) ⇔ x>− 1 ln 2 [≈ −0.23] andf 0 3 3
(x) < 0 if x 0 [the sum of two positive terms].
point of inflection.
Thus, f is concave upward on (−∞, ∞) and there is no
17. (a) y = f(x) = ln x √
x
.(Notethatf is only defined for x>0.)
f 0 (x) =
√
x (1/x) − ln x
1
2 x−1/2
x
=
1
√ − ln x
x 2 √ x
· 2 √ x
x 2 √ x = 2 − ln x > 0 ⇔ 2 − ln x>0 ⇔
2x 3/2
ln xe 8/3 ,sof is concave upward on (e 8/3 , ∞) and
concave downward on (0,e 8/3 ).Thereisaninflection point at
e 8/3 , 8 3 e−4/3 ≈ (14.39, 0.70).