(a) Prove that a polynomial function f of degree 1 has no extrema on the interval (-∞, ∞). (b) D

(a) Prove that a polynomial function f of degree 1 has no...

Key Concepts

Linear Functions

A linear function is a polynomial of degree one expressed in the form f(x) = mx + b. Its graph is a straight line, which means it has a constant rate of change and no curvature. This lack of curvature ensures that the function does not have any turning points, making it either strictly increasing or strictly decreasing over its entire domain.

Monotonicity

Monotonicity refers to the property of a function regarding its consistent direction of change. For a linear function, because the slope is constant (either positive or negative), it is strictly monotonic. This strict monotonic behavior indicates that the function does not change from increasing to decreasing or vice versa, which prevents the existence of any local extrema in an unbounded interval.

Critical Points and Extrema

Critical points of a function occur where its derivative is zero or undefined, and these points are potential candidates for local extrema. In the case of a linear function, the derivative is constant and does not vanish or become undefined (except for the trivial case of a constant function). As a result, there are no critical points to analyze for local maximum or minimum values within an open interval.

Extreme Value Theorem and Closed Intervals

The Extreme Value Theorem guarantees that any continuous function on a closed and bounded interval will achieve both a maximum and a minimum value. For a linear function defined on a closed interval [a, b], this means that even though the function does not exhibit any interior extrema due to its monotonicity, the absolute extrema must occur at one or both of the endpoints of the interval. Therefore, the analysis of extrema on a closed interval involves evaluating the function at the endpoints.

Transcript

00:01 Okay, so this problem wants us to look at a polynomial with degree 1.

00:06 So this is going to be a function that has an x with an exponent of 1 and no higher exponent.

00:14 It does have to have that exponent of 1, so it can't have no x's, but it could have some sort of number in front of the x.

00:22 So it could be like 2x or negative 5x or something like that.

00:25 The only number that can't be in front of the x is 0, because if the 0 is there, the x goes away and it's not a degree 1 anymore.

00:35 It can also have something added to that.

00:37 It could have some sort of constant, and that number, that n, can be anything.

00:42 So i write this generic polynomial mx plus n, and it's like the most generic polynomial degree 1 that we can possibly have.

00:50 So that's what i'm going to use for this problem.

00:52 The first part wants us to check the extrema on negative infinity to positive infinity.

00:59 This is an open interval.

01:01 It's got those round parentheses, so we don't need to check the endpoints, the negative infinity and the infinity.

01:07 The only thing we have to check is see if there's any extrema inside the interval, which we can check by taking that first derivative.

01:14 This is the first derivative test.

01:17 When we take the first derivative, we're left with just that coefficient in front of the x, the m.

01:23 And the first derivative test tells us to set the derivative equal to 0.

01:28 So we have 0 equals m, except for we already said that m can't be 0, otherwise it wouldn't be a first degree anymore.

01:37 And so we have a contradiction here, which means that there are no extrema.

01:43 There's no minimums or maximums on this interval, which makes sense because a polynomial degree 1 is just going to be a linear function.

01:54 So it's either going to be a straight line going kind of diagonally like that, or like this, or it could be horizontal like that.

02:01 So there's no kind of curves like mountains or valleys to make minimums or maximums.

02:06 It just kind of keeps going forever in either direction.

02:10 The second part of this problem wants us to find the extrema on this closed interval, a, b...

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