How do you determine whether the function #f(x)= sinx-cosx# is concave up or concave down and its intervals?

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Answer 1 · 2015-10-25T15:16:53

See the explanation section.

#f(x)= sinx-cosx#

#f'(x)= cosx+sinx#

#f''(x)= -sinx+cosx#

#f''(x) = 0# where #sinx = cos x# or #tanx=1#

This happens at #x=pi/4 + pik# for integer #k#.

For #pi/4 < x < (5pi)/4# we have #sinx > cos x# so #f''(x) <0# and the graph of #f# is concave down.

For #(-5pi)/4 < x < pi/4# we have #sinx < cos x# so #f''(x) > 0# and the graph of #f# is concave up.

Both sine and cosine are periodic with period #2pi#, so

on intervals of the form #(pi/4+2pik, (5pi)/4+2pik)#, where #k# is an integer, the graph of #f# is concave down.

on intervals of the form #((-5pi)/4+2pik, pi/4+2pik)#, where #k# is an integer, the graph of #f# is concave up.

There are, of course other ways to write the intervals.