The graph of function y =f(x) is shown at the bottom.

a) Estimate the intervals where the function is increasing.

b) Estimate the intervals where f'(x) < 0.

c) Estimate the coordinates of the critical points.

d) Estimate the equations of any vertical asymptotes.

e) . If x \geq -6, estimate the intervals where f'(x) < 0 and f''(x) > 0.

f) Identify a point of inflection, and state the approximate ordered pair for the point.

a) Determine the critical points of the function f(x) = 2x^4 -8x^3 - x^2 +6x.

b) Classify each critical point in part a.

Sketch the graph of a function with the following properties:

• There are local extrema at (-1, 7) and (3, 2).
• There is a point of inflection at (1, 4).
• The graph is concave down only when x <1.
• The x-intercept is -4 and the y-intercept is 6.

Check the function \displaystyle g(x) = \frac{x^2 + 7x + 10}{(x -3)(x +2)} for discontinuities. Conduct appropriate tests to determine if asymptotes exist at the discontinuity values. State the equations of any asymptotes and the domain of g(x).

Sketch ha graph of a function f with all of the following properties:

• The graph is increasing when x < -2 and when $- 2< x < 4$`.
• The grap his decreasing when x > 4.
• f'(-2) =0, f'(4) = 0
• The graph his concave down when x < -2 and when 3 < x < 9.
• The graph his concave up when -2 < x < 3 and when x > 9.

Use at least five curve-sketching techniques to explain how to sketch the graph of the function \displaystyle f(x) = \frac{2x + 10}{x^2 -9} . Sketch the graph on graph paper.

The function f(x) = x^3 + bx^2 +c has a critical point at (-2 , 6).

a) Find the constants b and c.

b) Sketch the graph of f(x) using only the critical points and the second derivative test.