Solutions to Problem Set 15

Math 310-01/02

10-18-2017

1. Show by specific counterexample that the following statements are false.

(a) "$\sqrt{x^4 + x^2 + 1} =
   \sqrt{x^4} + \sqrt{x^2} + \sqrt{1}$ for all $x \in
   \real$ ."

(b) "The square of a real number is greater than or equal to the original number."

(c) "If m and n are integers and 6 divides $mn$ , then either 6 divides m or 6 divides n."

(d) "The derivative of a product of two functions is equal to the product of the derivatives of the two functions."

(a) If $x = 1$ , $\sqrt{x^4 + x^2 + 1} = \sqrt{3}$ , while $\sqrt{x^4} + \sqrt{x^2} + \sqrt{1} = 1 + 1 + 1 = 3$ . Therefore, the result is false.

(b) If $x = \dfrac{1}{2}$ , then $x^2 = \dfrac{1}{4}$ , but $\dfrac{1}{4}$ is not greater than or equal to $\dfrac{1}{2}$ . Therefore, the result is false.

(c) If $m = 2$ and $n
   = 3$ , then 6 divides $m n = 6$ , but 6 does not divide either $m = 2$ or $n = 3$ . Therefore, the result is false.

(d) Let $f(x) = x^2$ and $g(x) = x^3$ . Then

$$\der {} x f(x) g(x) = \der {} x (x^2 \cdot x^3) = \der {} x x^5 = 5 x^4, \quad\hbox{but}\quad$$

$$\left(\der {} x f(x)\right)\left(\der {} x g(x)\right) = \left(\der {} x x^2\right)\left(\der {} x x^3\right) = (2 x)(3 x^2) = 6 x^3.$$

Therefore, the result is false.


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