Sketch the graph of a function that satisfies all of the given conditions.

Question

Sketch the graph of a function that satisfies all of the given conditions.

f(x) is odd (symmetric through the origin).

f'(2)=0, f'(x)>0 if -22, lim x->infinity f(x)=2, f''(x)<0 if 0 < x <4, f''(x)>0 if x>4.

(show the finished graph)

Answer 1

Here Is how to sketch the graph

Start at origin (0,0). Draw a short diagonal line “ / “ through the origin.

We know that for 0 < x < 2, f’(x) > 0. It means that f(2) > f(0).

Pick a value for f(2) which is larger than zero. I am going to use 6. Draw a point at (2, 6)

We also know that f’(2) = 0. Draw a short horizontal line “_ “ through the point (2, 6)

We also know that f’’ (x) changes sign at x = 4. Pick some value for f(4), for example 3. Draw a diagonal line “ \ “ through the point (4, 3).

We also know that f’’ (x) < 0 on the interval 0 < x < 4.

Pick values for f(1) and f(3), which are closer to f(2) than to f(0) and f(4) respectively. I am going to use

f(1) = 4 and f(3) = 5

Draw the points (1, 4) and (3, 5)

Finally draw a dotted line at y = 2. This is the horizontal asymptote. The graph will be approaching it from above.

Connect the dots and the lines. Your function should look like an upside down parabola (but not symmetrical) between x = 0 and x = 4 and then like a hyperbola at x > 4

f(x) is an odd function, so to draw the part of the graph where x < 0, flip all the signs. For example draw a horizontal line through (-2, -6), and the horizontal asymptote will be at y = -2

I hope it helps. Reach out if you have any questions