91Ó°ÊÓ

Suppose that \(f\) is a differentiable one-one function with a nowhere zero derivative and that \(f=F^{\prime}\). Let \(G(x)=x f^{-1}(x)-F\left(f^{-1}(x)\right)\). Prove that \(G^{\prime}(x)=f^{-1}(x) .\) (Disregarding details, this problem tells us a very interesting fact: if we know a function whose derivative is \(f\). then we also know one whose derivative is \(f^{-1}\). But how could anyone ever guess the function \(G\) ? Two different ways are outlined in Problems \(14-14\) and \(19-16\).

Suppose \(h\) is a function such that \(h^{\prime}(x)=\sin ^{2}(\sin (x+1))\) and \(h(0)=3\) Find (i) \(\quad\left(h^{-1}\right)^{\prime}(3)\) (ii) \(\quad\left(\beta^{-1}\right)^{\prime}(3),\) where \(\beta(x)=h(x+1)\)

(a) Prove that an increasing and a decreasing function intersect at most once. (b) Find two continuous increasing functions \(f\) and \(g\) such that \(f(x)=g(x)\) precisely when \(x\) is an integer. (c) Find a continuous increasing function \(f\) and a continuous decreasing function \(g,\) defined on \(R,\) which do not intersect at all.

(a) If \(f\) is a continuous function on \(\mathbf{R}\) and \(f=f^{-1}\), prove that there is at least one \(x\) such that \(f(x)=x .\) (What does the condition \(f=f^{-1}\) mean geometrically?) (b) Give several examples of continuous \(f\) such that \(f=f^{-1}\) and \(f(x)=x\) for exactly one \(x\). Hint: Try decreasing \(f\). and remember the geometric interpretation. One possibility is \(f(x)=-x\) (a) If \(f\) is a continuous function on \(\mathbf{R}\) and \(f=f^{-1}\), prove that there is at least one \(x\) such that \(f(x)=x .\) (What does the condition \(f=f^{-1}\) mean geometrically?) (b) Give several examples of continuous \(f\) such that \(f=f^{-1}\) and \(f(x)=x\) for exactly one \(x\). Hint: Try decreasing \(f\). and remember the geometric interpretation. One possibility is \(f(x)=-x\)

A finction \(f\) is nondecreasing if \(f(x) \leq f(y)\) whenever \(x

(a) Suppose that \(f(x)>0\) for all \(x,\) and that \(f\) is decreasing. Prove that there is a contimuous decreasing function \(g\) such that \(0