Questions from General Calculus

Q: Show that a cubic function (a third-degree polynomial)

Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three x-intercepts x1, x2, and x3, show that the x-coordinate of the inflection poin...

Q: For what values of c does the polynomial P(x)

For what values of c does the polynomial P(x) = x4 + cx3 + x2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as c d...

Q: Prove that if (c, f (c)) is a

Prove that if (c, f (c)) is a point of inflection of the graph of f and f ’’ exists in an open interval that contains c, then f ’’(c) = 0.

Q: Show that if f (x) = x4, then f

Show that if f (x) = x4, then f ’’(0) = 0, but (0, 0) is not an inflection point of the graph of f .

Q: Sketch the graph of a function that satisfies all of the given

Sketch the graph of a function that satisfies all of the given conditions. (a) f ‘(x) < 0 and f ‘‘(x) < 0 for all x (b) f ‘(x) > 0 and f ‘‘(x) > 0 for all x

Q: (a) Use the Product Rule twice to prove that if

(a) Use the Product Rule twice to prove that if f , g, and h are differentiable, then s (fgh)’ = f’ gh +fg’ h + fgh’ . (b) Takin...

Q: Show that the function g(x) = x |x

Show that the function g(x) = x |x | has an inflection point at (0, 0) but g ’’(0) does not exist.

Q: Suppose that f ’’’ is continuous and f ‘(c) =

Suppose that f ’’’ is continuous and f ‘(c) = f ’’(c) = 0, but f ’’’(c) > 0. Does f have a local maximum or minimum at c? Does f have a point of inflection at c?

Q: Suppose f is differentiable on an interval I and f ‘(x

Suppose f is differentiable on an interval I and f ‘(x) > 0 for all numbers x in I except for a single number c. Prove that f is increasing on the entire interval I.

Q: The three cases in the First Derivative Test cover the situations one

The three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions f, g, and h whose values at 0 are all 0 and, for...