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### 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.

1. $$z = 1$$
2. $$z = 5$$
3. $$z = 20$$

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a Show Solution

We 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 $$z = 20$$.