In order to find the extrema, we need to solve the equation
dxdf(x)=0(the derivative equals zero),
and the roots of this equation are the extrema of this function:
dxdf(x)=the first derivativecos(x)+cos(2x)=0Solve this equationThe roots of this equation
x1=−35πx2=−πx3=−3πx4=3πx5=πx6=35πThe values of the extrema at the points:
___
-5*pi 3*\/ 3
(-----, -------)
3 4
(-pi, 0)
___
-pi -3*\/ 3
(----, --------)
3 4
___
pi 3*\/ 3
(--, -------)
3 4
(pi, 0)
___
5*pi -3*\/ 3
(----, --------)
3 4
Intervals of increase and decrease of the function:Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=−3πx2=35πMaxima of the function at points:
x2=−35πx2=3πDecreasing at intervals
[35π,∞)Increasing at intervals
(−∞,−3π]∪[3π,35π]