Hello! This function is infinitely differential on entire real line, therefore we can use derivatives to determine its behavior.
Note that f(x) = sinx + cosx = sqrt(2) sin(x+pi/4).
The first derivative is f'(x) = sqrt(2) cos(x + pi/4). It is zero (inside [0, 2 pi] ) at x_1 = pi/4 and x_2 = pi/4 + pi = (5pi)/4.
This way the first derivative is positive (and f(x) increases) on (0, pi/4) and on ((5pi)/4, 2pi) and is negative (and f(x) decreases) on (pi/4, (5pi)/4). Thus f(x) has a local maximum at x_1 = pi/4 with the value f(pi/4) = sqrt(2) and the local minimum at x_2=(5pi)/4 with the value f(x_2)=-sqrt(2).
The second derivative is f''(x) = -sqrt(2) sin(x+pi/4), and it is negative on (0, (3pi)/4) and on ((7pi)/4,2pi) and is positive on ((3pi)/4,(7pi)/4). Therefore f(x) is concave down on (0, (3pi)/4) and on ((7pi)/4,2pi) and is concave up on ((3pi)/4,(7pi)/4).
The inflection points are where direction of concavity changes, i.e. (3pi)/4 and (7pi)/4.
Look at the picture made with desmos.com. The function is in green, the first derivative is dashed in red and the second derivative is dotted in red.
Tuesday, September 4, 2012
Find the intervals on which f is increasing or decreasing and find the local maximum and minimum values of f. Find the intervals of concavity and the inflection points. f(x)=sinx+cosx, 0
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