## Calculus: Early Transcendentals 8th Edition

The arrangement in increasing value is as follows: $$g'(0)\lt0\lt g'(4)\lt g'(2)\lt g'(-2)$$
Since the derivative of a number $a$ is shown in the graph as the tangent line of the curve $y=f(x)$ at the point $A(a, f(a))$, when you look at the curve, we can deduce some conclusions: 1) When the curve goes up, the tangent line at any point there would have a positive slope, so the derivative would be positive. 2) When the curve goes down, the tangent line at any point there would have a negative slope, so the derivative would be negative. 3) The steeper the curve goes up, the slope of the tangent line at a point there would have a higher value, so the derivative would consequently have a higher value. 4) The steeper the curve goes down, the slope of the tangent line at a point there would have a lower value, so the derivative would consequently have a lower value. Here for the function $g(x)$, we notice that in 4 points mentioned, only $x=0$ is where the curve goes down, so $g'(0)\lt0$ All the points $x=-2$, $x=2$ and $x=4$ all stay where the curve goes up, therefore, $g'(-2)\gt0$, $g'(2)\gt0$ and $g(4)\gt0$ However, we see that the curve is least steepest at point $x=4$ and most steepest at point $x=-2$. Therefore, $g'(4)\lt g'(2)\lt g'(-2)$. In conclusion, we have following arrangement: $$g'(0)\lt0\lt g'(4)\lt g'(2)\lt g'(-2)$$