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​Given f,g are continuous functions on [0,1] such that f(0)=f(1)=0, g(0)=g(1)=1, and f(1/2)>g(1/2
Question

Given f,g are continuous functions on [0,1] such that f(0)=f(1)=0, g(0)=g(1)=1, and f(1/2)>g(1/2).Which of the following statements is true?\text{Given } f, g \text{ are continuous functions on } [0, 1] \text{ such that } \\f(0) = f(1) = 0, \; g(0) = g(1) = 1, \; \text{and } f(1/2) > g(1/2). \\\text{Which of the following statements is true?}​​

A.

There is no, t[0,1]t\in [0,1] such that f(t)=g(t) .​

B.

there is exactly one t[0,1]t\in [0,1] such that f(t)=g(t) .​

C.

There are at least two t[0,1]t\in[0,1]​  such that f(t)=g(t) .

D.

There are always infinitely many t[0,1]t\in [0,1] such that f(t)=g(t) .​

Correct option is C

Given f,g are continuous functions on [0,1] such that f(0)=f(1)=0, g(0)=g(1)=1, and f(12)>g(12).Now, if f(12)>g(12), then the curve of f will intersect with the curve of g at least 2 times.This can be visualized with the help of rough diagrams of curves of f and g as follows:\text{Given } f, g \text{ are continuous functions on } [0, 1] \text{ such that } \\f(0) = f(1) = 0, \; g(0) = g(1) = 1, \; \text{and } f\left(\frac{1}{2}\right) > g\left(\frac{1}{2}\right). \\\text{Now, if } f\left(\frac{1}{2}\right) > g\left(\frac{1}{2}\right), \text{ then the curve of } f \text{ will intersect with the curve of } g \text{ at least 2 times.}\\\text{This can be visualized with the help of rough diagrams of curves of f and g as follows:}


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