What value of k ensures that the equation f(x)=6 has two solutions?

Question

x 0, 1, 2

f(x) 1, k, 2

Answer 1

I am assuming that f(x) is continuous on the interval [0,2].

 

Let f(1) = k, where k > 6.

 

Since f(x) is continuous on [0,1], f(0) < 6 and f(1) > 6, then by the Intermediate Value Theorem, there is at least one number, c₁, in the interval (0,1) such that f(c₁) = 6.

 

Since f(x) is continuous on [1,2], f(1)>6 and f(2)<6, then by the Intermediate Value Theorem, there is at least one number c₂ in the interval (1,2) such that f(c₂) = 6. 

 

So, choose k to be any number larger than 6.