the sign charts below give information for the first and second derivative of a function f. at which input…

the sign charts below give information for the first and second derivative of a function f. at which input below is the graph of f decreasing and concave down? f(x) + + + 0 - - - 0 + + + x -2 -1 0 1 2 f(x) - - - - - 0 + + + + + x -2 -1 0 1 2 x = -0.5 x = 1.5 x = 0 x = 0.5 x = -1.5

the sign charts below give information for the first and second derivative of a function f. at which input below is the graph of f decreasing and concave down? f(x) + + + 0 - - - 0 + + + x -2 -1 0 1 2 f(x) - - - - - 0 + + + + + x -2 -1 0 1 2 x = -0.5 x = 1.5 x = 0 x = 0.5 x = -1.5

Answer

Explanation:

Step1: Recall derivative - based properties

A function (y = f(x)) is decreasing when (f'(x)<0) and concave - down when (f''(x)<0).

Step2: Analyze the sign of (f'(x))

From the sign - chart of (f'(x)), (f'(x)<0) when (- 1<x<1).

Step3: Analyze the sign of (f''(x))

From the sign - chart of (f''(x)), (f''(x)<0) when (x < 0).

Step4: Find the intersection

We need to find the values of (x) that satisfy both (f'(x)<0) and (f''(x)<0). The intersection of (-1 < x<1) and (x < 0) is (-1<x<0).

Answer:

(x=-0.5)