Question

for the function shown in the figure below, at what labeled points is the slope of the graph positive? negative? at which labeled point does the graph have the greatest (i.e., most positive) slope? the least slope (i.e., negative and with the largest magnitude)?
o positive slope at d. negative slope at b and f. greatest slope at d. least slope at f.
o positive slope at c and d. negative slope at b and f. greatest slope at c. least slope at d.
o positive slope at a, c and d. negative slope at f. greatest slope at d. least slope at b.
o positive slope at a and d. negative slope at b, e and f. greatest slope at d. least slope at f.
o positive slope at a and d. negative slope at c and f. greatest slope at a. least slope at f.

Explanation:

Step1: Recall slope - positive/negative concept

The slope of a function \(y = f(x)\) at a point is the slope of the tangent line to the curve at that point. If the function is increasing at a point (going up as \(x\) increases), the slope of the tangent line is positive. If the function is decreasing at a point (going down as \(x\) increases), the slope of the tangent line is negative.

Step2: Analyze positive - slope points

At points \(A\) and \(D\), the function is increasing. So the slope of the graph is positive at \(A\) and \(D\).

Step3: Analyze negative - slope points

At points \(B\), \(E\), and \(F\), the function is decreasing. So the slope of the graph is negative at \(B\), \(E\), and \(F\).

Step4: Analyze greatest - slope point

The slope of the tangent line is the steepest (most positive) at point \(D\) among the given points.

Step5: Analyze least - slope point

The slope is most negative (has the largest magnitude among negative slopes) at point \(F\).

Answer:

Positive slope at \(A\) and \(D\). Negative slope at \(B\), \(E\) and \(F\). Greatest slope at \(D\). Least slope at \(F\).