What values of a and b make f(x)=x^3+a x^2+b x have a. a local maximum at x=-1 and a local minim

What values of a and b make f(x)=x^3+a x^2+b x have a. a...

Key Concepts

Systems of Equations in Parameter Determination

When a function contains unknown coefficients, using given conditions about the location and nature of extrema or inflection points leads to a system of equations. By setting the first and second derivatives equal to the necessary values at specified points, one can form equations that, when solved simultaneously, determine the values of these coefficients. This process is a common method in algebra and calculus for parameter fitting.

Critical Points

Critical points of a function occur where its first derivative is zero or undefined. These points are important because they can indicate the locations of local extrema (minima or maxima) and potential inflection points. Identifying and analyzing critical points is a fundamental step in understanding the behavior of a function.

First Derivative

The first derivative of a function provides the rate at which the function's value changes with respect to the independent variable. It is used to find critical points by setting it equal to zero, leading to equations that can be solved for unknown coefficients. This derivative is essential in determining where the slope of the function changes sign, which is indicative of local extrema.

Second Derivative and the Second Derivative Test

The second derivative of a function gives information about its concavity. By analyzing the sign of the second derivative at critical points, one can determine whether a critical point is a local maximum (if the second derivative is negative) or a local minimum (if it is positive). This test is a key tool in classifying the nature of critical points found from the first derivative.

Point of Inflection

A point of inflection is where a function changes its concavity, which is detected when the second derivative is zero and typically changes sign around that point. Recognizing points of inflection is important in understanding the overall shape and behavior of the function.

Transcript

00:01 Well, welcome.

00:01 We are going to talk about how to determine some values where local minimums and maximums occur.

00:11 And so we're starting with a function, f of x is equal to x cubed plus ax squared plus bx.

00:23 And the first thing we want to do is we want to find a and b where f of x has a local minimum at x equal to negative 1 and a local max at x equal to 3.

00:59 And we know that local minimums and maximums or relative minimum and maximums occur at critical numbers based off of where that first derivative equals zero.

01:09 So the first thing we're going to do is take the first derivative, and so that is 3x squared plus 2ax plus b.

01:21 Okay.

01:22 And so we're going to set that equal to zero.

01:25 And so we have 3x squared plus 2ax plus b.

01:32 Okay, so now we're told that we know that this function has a local minimum at negative 1.

01:42 And so we're in substituting negative 1 for x, and we get 0 is equal to 3 minus 2a plus b.

01:54 We also know that it has a relative maximum at three.

02:00 So we're going to do the same thing.

02:02 So zero is equal to 27 plus 6a plus b.

02:10 So now basically we have two equations and two unknowns.

02:16 And so we're going to add them together in such a way to find a or b.

02:23 And so here we have 2a minus b is equal to 3.