i need to solve this : -3^x +4^x +5^x <6^x

giorgiana1976 | Student

Therefore, we'll have to prove that `4^x-3^x lt 6^x - 5^x`

We'll use Lagrange's theorem to prove the inequality.

We'll choose a function f(x)=`t^x` , such as, if we'll differentiate it with respect to t, we'll get: `f'(x) = x*t^(x-1).`

According to Lagrange's theorem, applied over an interval [a,b],we'll get:

f(b) - f(a) = f'(c)(b-a)

We'll choose the intervals [3;4] and [5;6], such as:

`6^x - 5^x = x*c^(x-1)(6-5)`

`6^x - 5^x = x*c^(x-1)`

`4^x - 3^x = x*d^(x-1)(4-3)`

`` `4^x - 3^x = x*d^(x-1)`

We'll have to prove that `4^x - 3^x lt 6^x - 5^x =gt x*d^(x-1) ltx*c^(x-1)`

We'll reduce by x which is positive and it will keep the direction of the inequality unchanged:

`d^(x-1) lt c^(x-1)`

Since x is positive and d < c (d is in the interval [3;4] and c is in the interval [5;6]) => `d^(x-1)`  <`c^(x-1)`

According to Lagrange's theorem, the given inequality 4^x-`3^x lt 6^x - 5^x`  is verified.

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