Calculus
posted by Cynthia .
1. Locate the absolute extrema of the function f(x)=cos(pi*x) on the closed interval [0,1/2].
2. Determine whether Rolle's Theorem applied to the function f(x)=x^2+6x+8 on the closed interval[4,2]. If Rolle's Theorem can be applied, find all values of c in the open interval (4,2) such that f'(c)=0.
3. Determine whether the open intervals on which the graph of f(x)=7x+7cosx is concave upward or downward.
4. Find the points of inflection and discuss the concavity of the function f(x)=sinxcosx on the interval (0,2pi)
5.Find the points of inflection and discuss the concavity of the function f(x)=x^3+x^26x5

1.
f' = pi sin(pi*x) extrema where f' = 0, or x an integer
2.
since f(x) = (x+4)(x+2) f(4)=f(2)=0, so we're good to do. vertex is at x = 3.
3.
f is concave up if f'' > 0
f'' = 7cosx, so where is that >0? <0?
4.
concavity as above, inflection where f'' = 0
f'' = sinx + cosx = √2 sin(x + π/4)
5.
same methods as in #3,4/
f'' = 6x