Question

refer to the graph at the right to answer the following questions.
(a) at which labeled points is the function increasing?
(b) at which labeled points is the graph concave up?
(c) which labeled point has the most negative slope?
the function is increasing at the labeled point(s)
(use a comma to separate answers as needed.)
the graph is concave up at the labeled point(s)
(use a comma to separate answers as needed.)
the graph has the most negative slope at the labeled points(s)
(use a comma to separate answers as needed.)

Explanation:

Step1: Determine where the function is increasing

A function \(y = f(x)\) is increasing when the slope of the tangent line (derivative \(f^{\prime}(x)\)) is positive. Looking at the graph, as \(x\) increases, the \(y\) - value of the function is increasing at points \(A\) and \(F\).

Step2: Determine where the graph is concave up

A graph \(y = f(x)\) is concave up when the second - derivative \(f^{\prime\prime}(x)>0\). Geometrically, the graph “holds water” or the slope of the tangent line is increasing. At points \(D\), \(E\), \(F\), the graph is concave up.

Step3: Determine the point with the most negative slope

The slope of the tangent line at a point on the graph of \(y = f(x)\) is given by \(m = f^{\prime}(x)\). The most negative slope means the steepest downward - sloping tangent line. At point \(D\), the tangent line has the steepest downward slope among the labeled points.

Answer:

The function is increasing at the labeled point(s) \(A,F\).
The graph is concave up at the labeled point(s) \(D,E,F\).
The graph has the most negative slope at the labeled points(s) \(D\).