Question

Fill in the blanks to complete each of the following theorem statements.

Tests for Monotonicity: A sequence {ak} is increasing if it passes any of the following tests:

The Derivative Test: $\_\_\_\_\ge 0\mathrm{for}\mathrm{all}x\ge 1$ given that a(x) is a function that is _______on $[1,\infty )$ and whose value at any positive integer k is$a\left(k\right)=\_\_\_\_$

Short answer

The required answer is $a\text{'}\left(x\right)\ge 0$for all $x\ge 1$given that a(x) is a function that is differentiable on $[1,\infty )$and whose value at any positive integer k is$a\left(k\right)=a_k$

Step-by-step solution

Step 1. Given Information 

The given term is Derivative test.

Step 2. Explanation 

The derivative test is based on the test for increasing behavior in differentiable functions.

Thus,

$a\text{'}\left(x\right)\ge 0\mathrm{for}\mathrm{all}x\ge 1$given that a(x) is a function that is differentiable on $[1,\infty )$and whose value at any positive integer k is$a\left(k\right)=a_k$