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CSIR NET MATHEMATICAL SCIENCES

Real Analysis

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

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

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

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

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

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

Solution

Correct option is C

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