Question

Consider the function f(x)=4x3+2x2+5, and let c be a number in the interval [01]. For what values of k is there a c in this interval such that f(c)=k?

Answer 1

The Intermediate Value Theorem guarantees that there is a value c such that f(c) = k for 0≤c≤1 and f(0)≤k≤f(1), if f(1)≥f(0), and f(0)≥k≥f(1) if f(1)<f(0)."

See also reply to following post:
http://www.jiskha.com/display.cgi?id=1297024569

Answer 2

show that f satisfies the hypotheses of Rolle's theorem on [a,b],and find all number c in (a,b) such that f '(c)=0. f(x)=cos2x+2cosx ; [0,2(180)]