Question

For the function \(t\) whose graph is given, arrange the following numbers in increasing order and explain your reasoning:

\(0 \,\,g'\left( { - 2} \right) \,\,\,g'\left( 0 \right) \,\,\,g'\left( 2 \right) \,\,\,g'\left( 4 \right)\)

Short answer

The numbers in increasing order as follows: \(g'\left( 0 \right) < 0 < g'\left( 4 \right) < g'\left( 2 \right) < g'\left( { - 2} \right)\)

Step-by-step solution

Slope of a function at a point from the graph

We can determine the slope of a function at a point \(P\left( {a,f\left( a \right)} \right)\) from a graph will be \(m = \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) - f\left( a \right)}}{{x - a}}\). Also we can observe and compare the slope of the function of two points without calculating.

Comparing the slopes at some certain points from the graph 

From the graph we can notice that slope at the point \(0\) is negative and slope at point \( - 2\) will be the greatest. Comparing the slopes at \(2\) and \(4\) we can see that slope at point \(2\) is greater than that of at \(4\). Hence the increasing order is

\(g'\left( 0 \right) < 0 < g'\left( 4 \right) < g'\left( 2 \right) < g'\left( { - 2} \right)\).