Problem 107
Question
Use the definitions of increasing and decreasing functions to prove that \(f(x)=1 / x\) is decreasing on \((0, \infty)\).
Step-by-Step Solution
Verified Answer
By choosing any two arbitrary points 'a' and 'b' within the domain (0, \(\infty\)) and such that \(a < b\), and by evaluating the function at these points, it can be observed that \(f(a) > f(b)\). Hence, by definition, the function \(f(x) = 1 / x\) is decreasing for \(x > 0\).
1Step 1: Define the function and its domain
The function to be tested is \(f(x) = 1 / x\) and the domain that we need to consider is \(x\) greater than 0 (i.e., (0, \(\infty\))).
2Step 2: Choose arbitrary numbers within the domain
Select arbitrary numbers \'a\' and \'b\' in the domain such that \(0 < a < b\). Now, the task is to prove that whenever \(a < b\), \(f(a) > f(b)\). Let's compute the function values at these points and compare them.
3Step 3: Compute function values at chosen points
The value of the function \(f(x)\) at points \'a\' and \'b\' are \(f(a) = 1/a\) and \(f(b) = 1/b\) respectively.
4Step 4: Comparison of function values
Since a and b are positive and \(a < b\), it implies that \(1/a > 1/b\). Thus, it can be said that \(f(a) > f(b)\) whenever \(a < b\). Therefore, by definition, the function \(f(x) = 1 / x\) is decreasing on the domain (0, \(\infty\)).
Other exercises in this chapter
Problem 105
Prove that if \(p(x)=a_{n} x^{n}+\cdots+a_{1} x+a_{0}\) and \(q(x)=b_{m} x^{m}+\cdots+b_{1} x+b_{0}\left(a_{n} \neq 0, b_{m} \neq 0\right),\) then \(\lim _{x \r
View solution Problem 106
Use the definitions of increasing and decreasing functions to prove that \(f(x)=x^{3}\) is increasing on \((-\infty, \infty)\).
View solution Problem 105
Let \(x>0\) and \(n>1\) be real numbers. Prove that \((1+x)^{n}>1+n x\).
View solution