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Question Number 82421 by M±th+et£s last updated on 21/Feb/20

Answered by mind is power last updated on 21/Feb/20

$$lety\in [a,b]suchsupf=f\left(y\right)$$$$letz\in [a,b]suchsupg=g\left(z\right)$$$$ifz=y\Rightarrow c=y\Rightarrow f\left(c\right)=g\left(c\right)$$$$\text{You can't use 'macro parameter character \#' in math mode}$$$$leth\left(x\right)=f\left(x\right)-g\left(x\right)$$$$h\left(z\right)=f\left(z\right)-g\left(z\right)=supf-g\left(z\right)=supg-g\left(z\right)⩾0$$$$h\left(y\right)=f\left(y\right)-supg=f\left(y\right)-supf⩽0$$$$\Rightarrow h\left(z\right).h\left(y\right)⩽0$$$$sincehiscontinus\Rightarrow \mathrm{\exists }c\in [z,y]suchthat$$$$h\left(c\right)=f\left(c\right)-g\left(c\right)=0\iff f\left(c\right)=g\left(c\right)$$

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