f(t)=4cost+2sin2t, [0,pi/2]
What is ABSOLUTE MAX?

f(t)=4⋅cos(t)+2⋅sin(2t)0≤t≤π2f′(t)=4⋅[−sin(t)]+2⋅cos(2t)⋅2f′(t)=−4⋅sin(t)+4⋅cos(2t)|find min/max f′(t)=00=−4⋅sin(t)+4⋅cos(2t)|:40=−sin(t)+cos(2t)cos(2t)−sin(t)=0|cos(2t)=cos2(t)−sin2(t)|=1−sin2(t)−sin2(t)|=1−2sin2(t)cos(2t)−sin(t)=0|cos(2t)=1−2sin2(t)1−2sin2(t)−sin(t)=0|⋅(−1)−1+2sin2(t)+sin(t)=02sin2(t)+sin(t)−1=0|substitute: u=sin(t)2u2+u−1=0u1,2=−1±√(−1)2−4⋅2⋅(−1)2⋅2u1,2=−1±√1+84u1,2=−1±√94u1,2=−1±34u1=−1+34u1=12u2=−1−34u2=−1t1=arcsin(u1)t1=arcsin(12)t1=30∘ or t1=π6t2=arcsin(u2)t2=arcsin(−1)t2=−90∘ or t2=−π2|no solution because 0≤t≤π2
f(t)=4⋅cos(t)+2⋅sin(2t)f(30∘)=4⋅cos(30∘)+2⋅sin(2⋅30∘)f(30∘)=4⋅cos(30∘)+2⋅sin(60∘)f(30∘)=4⋅√32+2⋅√32f(30∘)=2⋅√3+√3f(30∘)=3⋅√3f(30∘)=5.19615242271
Max or min at t=30∘(=π6)f(30∘)=5.196…?
y"(t)=−4⋅cos(t)−4⋅sin(2t)⋅2y"(t)=−4⋅cos(t)−8⋅sin(2t)y"(30∘)=−4⋅cos(30∘)−8⋅sin(2⋅30∘)y"(30∘)=−4⋅cos(30∘)−8⋅sin(60∘)y"(30∘)=−4⋅√32−8⋅√32y"(30∘)=−12⋅√32y"(30∘)=−6⋅√3|maximum because y″(30∘)<0
Max at t=30∘(=π6)f(30∘)=5.196…
