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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