Question

Real Analysis: Suppose
f:R → R
and $f'(x)=0$ for all . Prove that there exists $c \in \Bbb R$ such that $f(x)=c$ for all . Thanks in advance!

Answer 1

Given if' (1) 20 AREIR.. We have to prove f (W) ze THEIR and for some CEiR le. &RYER; f() fly) hef is constant... Now, let us take two point X,YER such that nicy Then since fenist on IR then f is continuous on [x, y], and f is differentiable on (ni) Then by to Lagrange's Mean Value Theorem, some

JER such that, frus- f(y) +(3) s = . = Since and . x-y fra- dey). Tas '(W) ?0 VULEIRI nery fo- fey)=0 fas re-Y 0]. f(a) = f(y) ny are arbitry in R. Then fis constant in R. I some CER such that. foxrac HRER (Proved).