Analysis | Part IA, 2005

Prove that if ff is a continuous function on the interval [a,b][a, b] with f(a)<0<f(b)f(a)<0<f(b) then f(c)=0f(c)=0 for some c(a,b)c \in(a, b).

Let gg be a continuous function on [0,1][0,1] satisfying g(0)=g(1)g(0)=g(1). By considering the function f(x)=g(x+12)g(x)f(x)=g\left(x+\frac{1}{2}\right)-g(x) on [0,12]\left[0, \frac{1}{2}\right], show that g(c+12)=g(c)g\left(c+\frac{1}{2}\right)=g(c) for some c[0,12]c \in\left[0, \frac{1}{2}\right]. Show, more generally, that for any positive integer nn there exists a point cn[0,n1n]c_{n} \in\left[0, \frac{n-1}{n}\right] for which g(cn+1n)=g(cn)g\left(c_{n}+\frac{1}{n}\right)=g\left(c_{n}\right).

Typos? Please submit corrections to this page on GitHub.