sketch the graph of a function that has the properties described below. f( - 2)=3; f( - 2)=0; concave down…

sketch the graph of a function that has the properties described below. f( - 2)=3; f( - 2)=0; concave down for all x choose the correct graph below.

sketch the graph of a function that has the properties described below. f( - 2)=3; f( - 2)=0; concave down for all x choose the correct graph below.

Answer

Explanation:

Step1: Analyze given conditions

We know (f(-2)=3) means the point ((-2,3)) is on the graph. (f'(-2) = 0) means the slope of the tangent - line at (x=-2) is (0) (horizontal tangent). Concave down for all (x) means (f''(x)<0) for all (x), so the graph is curved down - ward everywhere.

Step2: Check each option

  • Option A: The point ((-2,3)) is on the graph, and it has a horizontal tangent at (x = - 2), and the graph is concave down.
  • Option B: The point on the graph is ((3,-2)) instead of ((-2,3)), so it's wrong.
  • Option C: The point on the graph is ((-2,0)) instead of ((-2,3)), so it's wrong.
  • Option D: The point on the graph is ((-2,0)) instead of ((-2,3)), so it's wrong.
  • Option E: The graph is concave up near (x=-2), so it's wrong.
  • Option F: The point on the graph is ((3,-2)) instead of ((-2,3)), so it's wrong.
  • Option G: The graph does not have a horizontal tangent at (x=-2), so it's wrong.
  • Option H: The graph does not have a horizontal tangent at (x=-2), so it's wrong.

Answer:

A.