Section 3.6 : Derivatives of Exponential and Logarithm Functions
10. Determine if \(G\left( z \right) = \left( {z - 6} \right)\ln \left( z \right)\) is increasing or decreasing at the following points.
- \(z = 1\)
- \(z = 5\)
- \(z = 20\)
Show All Solutions Hide All Solutions
a Show SolutionWe know that the derivative of the function will give us the rate of change for the function and so we’ll need that.
\[G'\left( z \right) = \ln \left( z \right) + \frac{{z - 6}}{z}\]Now, all we need to do is evaluate the derivative at the point in question. So,
\[G'\left( 1 \right) = \ln \left( 1 \right) - 5 = - 5 < 0\]\(G'\left( 1 \right) < 0\) and so the function must be decreasing at \(z = 1\).
b Show Solution
We found the derivative of the function in the first part so here all we need to do is the evaluation.
\[G'\left( 5 \right) = \ln \left( 5 \right) - \frac{1}{5} = 1.40944 > 0\]\(G'\left( 5 \right) > 0\) and so the function must be increasing at \(z = 5\).
c Show Solution
We found the derivative of the function in the first part so here all we need to do is the evaluation.
\[G'\left( {20} \right) = \ln \left( {20} \right) + \frac{7}{{10}} = 3.69573\]\(G'\left( {20} \right) > 0\) and so the function must be increasing at \(z = 20\).