True or false?

Algebra Level 3

Consider the function f ( x ) = ln ⁡ ( x + 1 ) f(x) = \ln(x+1) on the range 0 < x < 1 0 < x < 1 .

True or false: f ( x ) > x f(x) > x for all x x in this range.

True False

Denton Young by Denton Young , 47, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Let g ( x ) = ln ⁡ ( x + 1 ) − x ⇒ g ′ ( x ) = 1 x + 1 − 1 < 0 ∀ x ∈ ( 0 , 1 ) ⇒ g(x) = \ln (x +1) - x \Rightarrow g '(x) = \frac{1}{x+1} - 1 < 0 \quad \forall x \in ( 0,1) \Rightarrow g(x) is a strictly decreasing function and g ( 0 ) = 0 ⇒ \quad \text{ and } g(0) = 0 \Rightarrow f ( x ) < x ∀ x ∈ ( 0 , 1 ) f(x) < x \quad \forall x \in (0,1) for example g ( 0.5 ) = ln ⁡ ( 1.5 ) − 0.5 < 0 ⇒ f ( 0.5 ) = ln ⁡ ( 1.5 ) < 0.5 g(0.5) = \ln (1.5) - 0.5 < 0 \Rightarrow f(0.5) = \ln (1.5) < 0.5

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Denton Young
Feb 17, 2016

f(0.9) = ln(1.9) = 0.64

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...