919 views

### By use of calculus, show that x − ln(1 + x) is positive for all positive x.

Let f be a function defined on the positive real axis with values in R, such that f(x) = x - ln(1+x). Differentiating this function, we obtain, f'(x) = (x - ln(1+x))' = x' - ln'(1+x) = 1 - 1/(1+x). Since, x > 0, we have 1 + x > 1, and so 1/(1+x) < 1. So, 1 - 1/(1+ x) > 0. So, f'(x) > 0, for all positive x. So, by L'Hospital Rule, f(x) is strictly increasing. Thus, f(x) > lim f(x) when x -> 0 = 0 - ln (1+0) = 0. So, f(x) > 0, for all positive. x.

1 year ago

## Still stuck? Get one-to-one help from a personally interviewed subject specialist

#### 31 SUBJECT SPECIALISTS

£28 /hr

Degree: Mathematics (Masters) - Warwick University

Subjects offered:.STEP., Maths+ 1 more

.STEP.
Maths
Further Mathematics

“Premium tutor. First class graduate with teaching experience from a top Russell Group university. I deliver fun and relaxed lessons which achieve results!”

£28 /hr

Degree: Mathematics (Masters) - Durham University

Subjects offered:.STEP., Physics+ 2 more

.STEP.
Physics
Maths
Further Mathematics

“Hi, I study maths at Durham; I averaged 89% last year. I have exclusively 5-star reviews from over 50 reviews. I'll refund any tutorial if you are dissatisfied.”

£25 /hr

Degree: Mathematics (Bachelors) - Oxford, Worcester College University

Subjects offered:.STEP., Maths+ 3 more

.STEP.
Maths
Further Mathematics
.MAT.
-Oxbridge Preparation-

“University of Oxford recent maths graduate, offering tutoring for GCSE, A level maths students, as well as Oxbridge preparation.”

£25 /hr

Degree: Mathematics and Computer Science (Masters) - Oxford, Merton College University

Subjects offered:.STEP., Maths+ 1 more

.STEP.
Maths
.MAT.

MyTutor guarantee

### You may also like...

#### Posts by Andreea

By use of calculus, show that x − ln(1 + x) is positive for all positive x.

How many distinct real roots does the equation x^3 − 30x^2 + 108x − 104 = 0 have?

The quadratic equation 2x^2 + 8x + 1 = 0 has roots a and b. Write down the value of a + b, a*b and a^2 + b^2.

#### Other Uni Admissions Test .STEP. questions

Given that a and b are distinct positive numbers, find a polynomial P(x) such that the derivative of f(x) = P(x)e^(−x²) is zero for x = 0, x = ±a and x = ±b, but for no other values of x.

By use of calculus, show that x − ln(1 + x) is positive for all positive x.

Prove: If pq, or p + q is irrational, then at least one of p and q is irrational.

Find all positive integers n such that 12n-119 and 75n-539 are both perfect squares. Let N be the sum of all possible values of n. Find N.

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this.