Answer to Question #124164 in Calculus for joe chegg

Question

Sketch the graph of a continuous function f(x) satisfying the following properties:
(i) the graph of f goes through the origin
(ii) f'(−2) = 0 and f'(3) = 0.
(iii) f'(x) > 0 on the intervals (−∞, −2) and (−2, 3).
(iv) f'(x) < 0 on the interval (3, ∞).
Label all important points.

Answer 1

Let's analyze the function properties:

(i) the graph of f goes through the origin means that the point (0, 0) is on the graph of f;

(ii) f'(−2) = 0 and f'(3) = 0 means that x=-2 and x=3 are critical points of f;

(iii) f'(x) > 0 on the intervals (−∞, −2) and (−2, 3) means that our function increases on both intervals and doesn't have an extremum point at x= -2, but at this point f has the horizontal tangent line (-2 is a point of inflection).

(iv) f'(x) < 0 on the interval (3, ∞) means that f decreases after 3, and as f increases before 3 and f is defined at x=3, so f has a relative maximum point at x=3.

Now we can sketch the graph of f: