Find the critical points of f ( x ) = cos 2 x + sin x in the interval (? ? , ? )

Question

Find the critical points of

f(x) = cos2x + sin x

in the interval (??, ?) and apply the Second Derivative Test (if possible) to determine whether each corresponds to a local minimum or maximum. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Find the critical points of rx) cos2 x + sin x in the interval (-, ) and apply the Second Derivative Test (if possible) to determine whether each corresponds to a local minimum or maximum. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) becil minima local minima x=12, 2 local maxima x- 0

Explanation / Answer

given f(x) = cos2x + sinx

f'(x) = 2cosx*(-sinx) + cosx

for critical points f'(x) = 0

2cosx*(-sinx) + cosx = 0

cosx(1-2sinx) = 0

cosx = 0 and 1-2sinx =0

x =pi/2, sinx = 1/2

x = pi/6

now f"(x) = - 2cos2x - sinx

f"(pi/2) = - 2cospi - sinpi/2 = 1 > 0 ( local minima

f"(pi/6) = --2cospi/3 - sinpi/6 = -3/2 < 0, (local maxima

  

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.