(a) Find all critical points of f(x)=x^3-a x+b (b) Under...

(a) Find all critical points of $f(x)=x^{3}-a x+b$ (b) Under what conditions on $a$ and $b$ does this function have no critical points? (c) Under what conditions on $a$ and $b$ does this function have exactly one critical point? What is the one critical point, and is it a local maximum, a local minimum, or neither? (d) Under what conditions on $a$ and $b$ does this function have exactly two critical points? What are they? Which are local maxima, which are local minima, and which are neither? (e) Is it ever possible for this function to have more than two critical points? Explain.

(a) Find all critical points of $f(x)=x^{3}-a x+b$
(b) Under what conditions on $a$ and $b$ does this function have no critical points?
(c) Under what conditions on $a$ and $b$ does this function have exactly one critical point? What is the one critical point, and is it a local maximum, a local minimum, or neither?
(d) Under what conditions on $a$ and $b$ does this function have exactly two critical points? What are they? Which are local maxima, which are local minima, and which are neither?
(e) Is it ever possible for this function to have more than two critical points? Explain.

Key Concepts

Discriminant Analysis in Quadratic Equations

When the first derivative of a function results in a quadratic equation, the discriminant (the expression under the square root in the quadratic formula) plays a key role in determining the number of real solutions. The sign of the discriminant indicates whether there are two distinct real solutions, one repeated real solution, or no real solutions, which in turn informs the number of critical points a function has.

Second Derivative Test

The second derivative test is used to classify the nature of critical points found from the first derivative. By computing the second derivative at a critical point, one can determine whether the point is a local minimum (if the second derivative is positive), a local maximum (if the second derivative is negative), or inconclusive (if the second derivative is zero). This method is widely used in optimization problems and curve analysis.

Inflection Points

An inflection point is where the function changes its concavity, meaning it shifts from being concave upward to concave downward or vice versa. While inflection points can sometimes coincide with critical points, they are specifically characterized by a change in the sign of the second derivative. In some cases, a critical point that yields a zero second derivative may indicate an inflection point rather than a local extremum.

Critical Points

Critical points of a function are those values in the domain where the derivative is zero or undefined. They are central to the study of calculus as they often correspond to local extrema or points of inflection. Identifying these points helps in understanding the behavior of the function, such as where it might change from increasing to decreasing or vice versa.

Derivative

The derivative represents the instantaneous rate of change of a function and is a foundational tool in calculus. By taking the derivative of a function and setting it equal to zero, one can solve for the values where the function’s slope is horizontal. These solutions are potential locations for local maximums, minimums, or inflection points, making the derivative critical in analyzing function behavior.

Transcript

Please give Ace some feedback