### Home > PC3 > Chapter 13 > Lesson 13.3.3 > Problem13-127

13-127.

Let $g(x)=3x+\frac{12}{x}$ for $x>0$.

1. Sketch a graph of $y = g\left(x\right)$, showing that the minimum value of the function occurs when $x = 2$.

2. Calculate the slope of the line tangent to $y = g\left(x\right)$ when $x = 2$.

$g^\prime(2)= \lim \limits_{h\to0}\frac{g(2+h)-g(2)}{h}$

$\lim \limits_{h\to0}\frac{3(2+h)+\frac{12}{2+h}-\left(3(2)+\frac{12}{2}\right)}{h}$

$\lim \limits_{h\to0}\frac{6+3h+\frac{12}{2+h}-12}{h}$

$\lim \limits_{h\to0}\frac{-6+3h+\frac{12}{2+h}}{h}\cdot \frac{(2+h)}{(2+h)}$