Ok, so remember that the derivitive of the position function is the velocty function and the derivitive of the velocity function is the accceleration function
x(t) is the positon function so just take the derivitive of 3t/π +cos(t) twice first derivitive is 3/π-sin(t) 2nd derivitive is -cos(t) a(t)=-cos(t)
on the interval [π/2,5π/2) where does -cos(t)=1? or where does cos(t)=-1? at t=π so now plug that in for t in the position function to find the position at time t=π x(π)=3(π)/π+cos(π) x(π)=3-1 x(π)=2 so the position is 2
ok, that graph is the first derivitive of f(x) the function f(x) is increaseing when the slope is positive it is concave up when the 2nd derivitive of f(x) is positive
we are given f'(x), the derivitive of f(x) we want to find where it is increasing AND where it is concave down
it is increasing when the derivitive is positive, so just find where the graph is positive (that's about from -2 to 4)
it is concave down when the second derivitive (aka derivitive of the first derivitive aka slope of the first derivitive) is negative where is the slope negative? from about x=0 to x=2 and that's in our range of being increasing so the interval is (0,2)
