consider the graph of the original function f(x) shown below. at x = -1, we know that f(x) is select and…

consider the graph of the original function f(x) shown below. at x = -1, we know that f(x) is select and f(x) is select.

consider the graph of the original function f(x) shown below. at x = -1, we know that f(x) is select and f(x) is select.

Answer

Explanation:

Step1: Analyze the slope at (x = - 1)

The slope of the tangent - line to the function (y = f(x)) at (x=-1) is negative. Since (f^{\prime}(x)) represents the slope of the tangent - line to the curve (y = f(x)) at the point (x), when (x = - 1), (f^{\prime}(x)<0).

Step2: Analyze the concavity at (x=-1)

The graph of (y = f(x)) is concave - up at (x=-1). The second - derivative (f^{\prime\prime}(x)) determines the concavity of the function. If the function is concave - up, then (f^{\prime\prime}(x)>0).

Answer:

(f^{\prime}(x)) is negative and (f^{\prime\prime}(x)) is positive.