Intervals in which f(x) is Strictly Increasing/DecreasingFind the intervals in which f ( x ) = sin ⁡ x + cos ⁡ x , 0 ≤ x ≤ 2 π is strictly increasing/decreasing.First I find the derivative f ′ ( x ) = cos ⁡ x − sin ⁡ x, then put f ′ ( x ) = 0, getting tan ⁡ x = 1. The principal solutions of tan ⁡ x = 1 are x = π / 4 and x = 5 π / 4, which gives the intervals [ 0 , π / 4 ), ( π / 4 , 5 π / 4 ), and ( 5 π / 4 , 2 π ).After this I am stuck. The book just solves the question by making a table showing the interval, sign of the derivative as positive or negative, and strictly increasing for positive and vice versa.I just want to know how do they get it positive or negative.It would really help if someone did one for the interval ( π / 4 , 5 π / 4 ).

Question

Intervals in which f(x) is Strictly Increasing/DecreasingFind the intervals in which   f  (  x  )  =  sin  ⁡  x  +  cos  ⁡  x  ,  0  ≤  x  ≤  2  π is strictly increasing/decreasing.First I find the derivative       f    ′    (  x  )  =  cos  ⁡  x  −  sin  ⁡  x, then put       f    ′    (  x  )  =  0, getting   tan  ⁡  x  =  1. The principal solutions of   tan  ⁡  x  =  1 are   x  =  π      /    4 and   x  =  5  π      /    4, which gives the intervals   [  0  ,  π      /    4  ),   (  π      /    4  ,  5  π      /    4  ), and   (  5  π      /    4  ,  2  π  ).After this I am stuck. The book just solves the question by making a table showing the interval, sign of the derivative as positive or negative, and strictly increasing for positive and vice versa.I just want to know how do they get it positive or negative.It would really help if someone did one for the interval   (  π      /    4  ,  5  π      /    4  ).

Taxinov · Accepted Answer

Step 1The function f′(x) is continuous in   [  0  ,  2  π  ]. We have       f    ′    (  x  )  =  0 at   x  =      π    4   and   x  =            5      π        4  .Then, it can be concluded that f′(x) is either positive for all x in   (      π    4    ,            5      π        4    ) or negative for all x in   (      π    4    ,            5      π        4    ). This follows from the continuity of f′(x) and the intermediate value theorem.Step 2Now, all that is left to find is whether it takes positive or negative values. This can be done easily by checking the value of f′(x) at any convenient x in the interval.For example, at   x  =      π    2  ,       f    ′    (  x  )  =  −  1.Thus, f′(x) is negative in   (      π    4    ,            5      π        4    ) and it can be concluded that f(x) is strictly decreasing in this interval.

Find the intervals in which f(x)=sin x+ cos x, 0 leq x leq 2pi is strictly increasing/decreasing.

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