Question

Use the Mean Value Theorem to show that  sqrt(1 + x) < 1 + (1/ 2)x if x > 0. (Hint: Apply the Mean Value Theorem to the function

f(x) = sqrt(1 + x))

Answer 1

Step 1:

The function is  .

Mean value theorem :

Let f be a function that satisfies the following three hypotheses :

1. f is continuous on .

2. f is differentiable on .

Then there is a number c in such that  .

Step 2:

Let x > 0 , .

Let us consider a number c in .

Since the hypotheses of the Mean Value Theorem are satisfied,

we get  .

Differentiate  with respect to .

Answer 2

Contd.....

Step 3:

Substitute in the mean value theorem.

If x > 0, then c > 0 and, therefore, .

So .

Solution: