Suppose f(x) & g(x) are continuous on [a,b] such that f(a) > g(a) and g(b) > f(b). What is the minimum number of solutions to f(x)=g(x) for x∈[a,b] ?

calculus

1 Answer
Andrea S.
Jan 31, 2018

#f(x) = g(x)# at least in one point in the interval #(a,b)#

Explanation:

Consider the function #h(x) = g(x) -f(x)#: as it is the sum of two continuous functions, also #h(x)# is continuous over the interval #[a,b]# and besides:

#f(a) > g(a) => h(a) < 0#

#f(b) < g(b) => h(b) > 0#

Since #h(x)# is continuous for #x in [a,b]# and has opposite signs at the limits of the interval, Bolzano's theorem states that there is at least one point in the interval #bar x in (a,b)# such that:

#h(bar x) = 0#

which means that:

#g(barx) - f(barx) =0#

and then:

#f(bar x ) = g(barx)#

Related questions
Impact of this question
518 views around the world
You can reuse this answer
Creative Commons License