Solution
We are given the function
[tex]P(x)=ln(4x+1)+3x-x^2[/tex]
We want to find how fast is the profit increasing when there are only 900 employees (x = 0.9).
To do that, we will have to differentiate P(x) with respect to x and then substitute the value of x = 0.9
Note
[tex]\begin{gathered} Given=ln(4x+1) \\ Differentiate=\frac{4}{4x+1} \\ \\ Given=3x \\ Differentiate=3 \\ \\ Given=x^2 \\ Differentiate=2x \end{gathered}[/tex]
Now, we differentiate
[tex]\begin{gathered} P(x)=ln(4x+1)+3x-x^{2} \\ B\text{y differentiating} \\ P^{\prime}(x)=\frac{4}{4x+1}+3-2x \\ \\ Pu\text{t x = 0.9} \\ \\ P^{\prime}(0.9)=\frac{4}{4(0.9)+1}+3-2(0.9) \\ \\ P^{\prime}(0.9)=\frac{238}{115} \\ \\ P^{\prime}(0.9)=2.069565217 \\ \\ P^{\prime}(0.9)=2.07million\text{ }dollars\text{ }per\text{ }thosandemployees \end{gathered}[/tex]
Therefore, the answer is
[tex]2.07million\text{ }dollars\text{ }per\text{ }thosand\text{ }employees[/tex]
