Question

Use the intermediate value theorem to prove that x(x-5)^2=5 has a solution between 4 and 6.

Answer 1

Solution :- the intermediate value theorem :- Suppose that f is a function continuous on a closed interval [a, b] and that f (a) and  f (b). If γ is some number between f (a) and f (b) then there must be at least one c : a < c < b for which f (c) = γ.

Now take f(x) = x(x-5)^2 -5 =0

here a = 4 and b = 6

then f(4) = 4(4-5)^2-5 =-1 and f(6) = 4(4-6)^2-5 = 11

clearly f(4) and f(6) are exist and finite . also f(4) and f(6) does not equals to each other .

hence by intermediate value theorem by the intermediate value theorem, there is at least one number c, in [4,6], for which f(c) = 0.