Fundamental theorem of calculus: Local max of f(t)?

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How would I go about finding the value of t in the local max of f(t)?

1 Answer
Andrea S.
May 12, 2018

#f(t)# has a local maximum in #t=-12#

Explanation:

Identify the critical points of the function solving the equation:

#f'(t) = 0#

Based on the fundamental theorem of calculus:

#f'(t) = (t^2+14t+24)/(1+cos^2t)#

As the denominator is always positive we have to solve the equation:

#t^2+14t+24 = 0#

#t= -7 +- sqrt (49 -24) = {(-2),(-12):}#

As the numerator is a second order polynomial with leading positive coefficients we know that it is negative inside the interval between the roots and positive outside. The denominator as we noted is always positive, so we can conclude that:

#f'(t) > 0 # for #t in (-oo,-12)#

#f'(-12) = 0#

#f'(t) < 0 # for #t in (-12,-2)#

#f'(-2) = 0#

#f'(t) > 0 # for #t in (-2,+oo)#

We can conclude that #f(t)# is monotone increasing from #-oo# to #t= -12# where it has a local maximum, monotone decreasing from #t=-12# to #t =-2# where it has then a local minimum, and again monotone increasing from #t=-2# to #+oo#.

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