Question

match the graphs a - d of functions with graphs a - c of their derivatives. the graph of function a corresponds to the graph of derivative the graph of function b corresponds to the graph of derivative the graph of function c corresponds to the graph of derivative the graph of function d corresponds to the graph of derivative

Explanation:

Step1: Recall derivative - slope relationship

The derivative of a function at a point is the slope of the tangent line to the function at that point.

Step2: Analyze function a

Function a is a decreasing linear - function. Its slope is negative and constant. So, its derivative is a negative constant function, which corresponds to graph C.

Step3: Analyze function b

Function b is an increasing linear - function. Its slope is positive and constant. So, its derivative is a positive constant function, which corresponds to graph A.

Step4: Analyze function c

Function c is a horizontal line. Its slope is 0. So, its derivative is the zero - function, which corresponds to graph B.

Step5: Analyze function d

Since functions a, b, and c are already matched, function d must correspond to the remaining graph.

Answer:

The graph of function a corresponds to the graph of derivative C.
The graph of function b corresponds to the graph of derivative A.
The graph of function c corresponds to the graph of derivative B.
The graph of function d corresponds to the remaining graph (not labeled in the problem statement but by elimination).