# Show that if two maximal values are equal on continuous functions, then there exists ψ∈[a,b]\psi \in [a,b] with f(ψ)=g(ψ)f(\psi) = g(\psi)

Let f,g:[a,b]→Rf,g : [a,b] \rightarrow \mathbb{R} be continuous. We know that ff and gg have maximal values, as they are continuous on a closed interval. Let MfM_f be the maximal value of ff, and MgM_g the maximal value of gg. Show that if MfM_f = MgM_g, then there exists ψ∈[a,b]\psi \in [a,b] with f(ψ)=g(ψ)f(\psi) = g(\psi)

Would it suffice to show that ψ\psi = maximal values, and show that this is an example which shows the exist of such a ψ\psi?

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If they take on their maximal values at the same point then we are done. If this is not the case, then use IVT on f−gf-g.
2 days ago

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