Find the local maxima and local minima, if any of the following function. Also, find the local maximum and the local minimum values, as the case may be as follows.
f(x)=sinx−cosx,0<x<2π
Find the local maxima and local minima, if any of the following function. Also, find the local maximum and the local minimum values, as the case may be as follows.
f(x)=sinx−cosx,0<x<2π
Given function is, f(x)=sinx−cosx, 0<x<2π
∴f′(x)=cosx+sinx and f"(x)=sinx+cosx
For maxima put f'(x)=0
⇒cosx+sinx=0⇒sinxcosx=−1, tanx=−1⇒x=π−π4,2π−π4⇒x=3π4,7π4,xϵ(2,2π)
∴ The points at which extremum may occur are 3π4 and 7π4
At x=π4,f"(3π4)=−sin3π4+cos3π4=−sin(π−π4)+cos(π−π4)
[∵sin(π−θ)=sinθ and cos(π−θ)=−cosθ]
=−sinπ4−cosπ4=−1√2−1√2=−√2<0
∴x=3π4 is a point of maxima.
Maxima value =f(3π4)=sin3π4−cos3π4
=sin(π−π4)−cos(π−π4)=sinπ4−cosπ4=1√2+1√2=2√2=√2
At x=7π4,f"(7π4)+cos7π4=−sin(2π−π4)+cos(2π−π4)
[∵sin(2π−θ)=−sinθ and cos(2π−θ)=cosθ]=sinπ4+cosπ4=1√2+1√2=√2>0
∴x=7π4 is a point of minima.
Minimum value
f(7π4)=sin7π4−cos7π4=sin(2π−π4)−cos(2π−π4)=−sinπ4−cosπ4=−1√2−1√2=−√2
Given function is, f(x)=sinx−cosx, 0<x<2π
∴f′(x)=cosx+sinx and f"(x)=sinx+cosx
For maxima put f'(x)=0
⇒cosx+sinx=0⇒sinxcosx=−1, tanx=−1⇒x=π−π4,2π−π4⇒x=3π4,7π4,xϵ(2,2π)
∴ The points at which extremum may occur are 3π4 and 7π4
At x=π4,f"(3π4)=−sin3π4+cos3π4=−sin(π−π4)+cos(π−π4)
[∵sin(π−θ)=sinθ and cos(π−θ)=−cosθ]
=−sinπ4−cosπ4=−1√2−1√2=−√2<0
∴x=3π4 is a point of maxima.
Maxima value =f(3π4)=sin3π4−cos3π4
=sin(π−π4)−cos(π−π4)=sinπ4−cosπ4=1√2+1√2=2√2=√2
At x=7π4,f"(7π4)+cos7π4=−sin(2π−π4)+cos(2π−π4)
[∵sin(2π−θ)=−sinθ and cos(2π−θ)=cosθ]=sinπ4+cosπ4=1√2+1√2=√2>0
∴x=7π4 is a point of minima.
Minimum value
f(7π4)=sin7π4−cos7π4=sin(2π−π4)−cos(2π−π4)=−sinπ4−cosπ4=−1√2−1√2=−√2
