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.
– basket
2 days ago

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2 Answers
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Suppose f(x1)=Mff(x_1) = M_f and g(x2)=Mf=Mgg(x_2) = M_f = M_g.

If x1=x2x_1 = x_2, you’re done.

Otherwise, consider the interval [x1,x2][x_1, x_2] (or [x2,x1][x_2, x_1] if x2