Find the critical numbers of the function f(\theta)=2\cos \theta+\sin^2 \thet

Tammy Todd

Tammy Todd

Answered question

2021-09-13

Find the critical numbers of the function f(θ)=2cosθ+sin2θ

Answer & Explanation

Nathaniel Kramer

Nathaniel Kramer

Skilled2021-09-14Added 78 answers

f(θ)=2cosθ+sin2θ
Because it is the sum of two differentiable functions,  f(θ) is differentiable.
As a result, the critical numbers are the equation's solutions f(θ)=0
Differentiate f(θ), To get
f(θ)=2sinθ+2sinθcosθ
f(θ)=2sinθ[cosθ1]
CASE 1: sinθ=0
Transform this equation into
sinθ=sin(0)
The general solution of the equation of the form sinθ=sinα is
θ=nπ+(1)nα
Consequently, the usual answer is
θ=nπ+(1)n(0)=nπ
CASE 2: cosθ=1
Rewrite this equation as
cosθ=cos(0)
The general solution of the equation of the form cosθ=cosα is
θ=2nπ±α
Therefore the general solution is
θ=2nπ±0=2nπ
Which is the subset of the solution from Case 1
Answer:
All numbers of the form θ=nπ are critical numbers

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