At what value of t does the local max occur?

Question

f(t)= integral (x^2+11x+24)/(1+cos^2(x)) from 0 to t 

At what value of t does the local max of f(t) occur?

Answer 1

By the Fundamental Theorem of Calculus,

   f'(t) = (t² + 11t + 24)/(1 + cos²t) = (t + 8)(t + 3)/(1 + cos²t)

 

   f'(t) = 0 when t = -3 or t = -8

 

When t < -8, f'(t) > 0.  So, f(t) is increasing when t < -8.

When -8 < t < -3, f'(t) < 0.  So, f(t) is decreasing when -8 < t < -3.

When t > -3, f'(t) > 0.  So, f(t) is increasing when t > -3.

 

Since f'(t) changes sign from positive to negative at t = -8, there is a relative maximum at t = -8.