The function f is continuous on the closed interval [0,2] and has values that are given in the table below

Question

x = 0| 1 | 2
____________
f(x) = 1| k | 2

The equation f(x) = 1/2 must have at least two solutions in the interval [0,2] if K = ?

The answer is 0, but there is also a choice option of 1/2. I don't understand how to get the answer.

Thanks!

Answer 1

For f(x)=1/2 to have two solutions on the interval [0,2], a horizontal line through y=1/2 must intersect f(x) two times or more on the interval.

This will happen if k<=1/2. In the limit where k=1/2, the horizontal line will be tangent to f(x), and is considered to have two solutions.