f(θ)=sin⁡θcos2θ−cot⁡θθ+1 Domain: [0,2π] Find: 1) what are the inflection point 2) relationship of stationary point and critical point 3) what are the critical points

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f(θ)=sin⁡θcos2θ−cot⁡θθ+1  Domain: [0,2π]  Find:  1) what are the inflection point  2) relationship of stationary point and critical point   3) what are the critical points

timbalemX · Accepted Answer

1) An inflection point is where the curve of the graph goes from concave down to up or vice versa. Point of inflection occur at f&quot;=02) Relationship of stationary point and critical point. We say x0 is a stationary point of a function if f(x) and f&#039;(x) exist and is equal to f′(x0)=0.And, x0 is a critical point of a function of f(x) if f(x0) exists and either f′(x0) does not exist (i.e. function is not differentiable at or f′(x0)=0.All stationary points are critical points but not all critical points are stationary points. 3) Point x0 is a critical point of a function of f(x) if f(x0) exists and either f′(x0) does not exist (i.e. function is not differentiable at x0 or f′(x0)=0.The first derivative test is a method of analyzing functions using their first derivatives in order to find their extremum point. We take the derivative of the function,f′(θ)=ddθ⋅(sin⁡θcos2θ−cos⁡θθ+1)=ddθ⋅(sin⁡θcos2θ)−ddθ(cos⁡θθ)+ddθ(1)=cos3θ−−θcos⁡ec2θ−cot⁡θθ2−sin⁡2θsin⁡θNow, substitute θ=4f′(0)=−0.27927−(−0.49047)−(−0.74875)=0.959951=∼0.96

f(theta)=sin theta cos^2 theta - (cot theta)/ theta + 1 Domain: [0, 2pi] Find: 1) what are the inflection point 2) relationship of stationary point and critical point 3) what are the critical points

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