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Chapter 4: Q. 74 (page 401)

Prove that lnxis increasing and concave down on its entire domain (0,鈭�).

Short Answer

Expert verified

lnx is increasing and concave down on its entire domain 0,鈭�.

Step by step solution

01

Step 1. Given information

We have to prove that lnx is increasing and concave down on its entire domain 0,鈭�.

02

Step 2. Proof of the question.

Consider the graph,

From the graph it is clear that the signed area under the graph of f=1tand x-axis is positive only on 0,鈭�.

Since, lnxis the antiderivative of 1twhere tis a dummy variable.

So,

ddxlnx=1x

Hence, for x>0derivative of lnxis positive so it is increasing.

Finding the second derivative,

d2dx2lnx=ddx1x=-1x2

-1x2is less than 0so the function is concave down.

Therefore, lnxis increasing and concave down on its entire domain 0,鈭�.

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Most popular questions from this chapter

Prove that 鈭�13(3x+4)dx=20in three different ways:

(a) algebraically, by calculating a limit of Riemann sums;

(b) geometrically, by recognizing the region in question as a trapezoid and calculating its area;

(c) with formulas, by using properties and formulas of definite integrals.

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

鈭�13(x+1)2dx

Show by exhibiting a counterexample that, in general, 鈭�f(x)g(x)dx鈮�鈭�f(x)dx鈭�g(x)dx. In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.

Shade in the regions between the two functions shown here on the intervals (a) [鈭�2, 3]; (b) [鈭�1, 2]; and (c) [1, 3]. Which of these regions has the largest area? The smallest?

Consider the general sigma notation 鈭�k=mnak.What do we mean when we say that ak is a function of k?
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