If an even function f(x) has a local maximum value at x=c, can anything be said about the value

step 1 So these are some true and false and multiple choice questions. I'm just going to do them all in

If an even function f(x) has a local maximum value at x=c,...

Key Concepts

Even Functions

An even function is one that satisfies the condition f(x) = f(-x) for all x in its domain. This property indicates that the graph of the function is symmetric with respect to the y-axis, meaning that for every value x, the function value at -x is exactly the same as at x.

Local Maximum

A local maximum is a point where the function value is greater than or equal to the values of the function at nearby points. This means that within some open interval around that point, no other point yields a higher function value than the local maximum.

Symmetry and Implications for Critical Points

Because even functions are symmetric about the y-axis, any feature of the function that occurs at a point c will have a corresponding feature at -c. Therefore, if there is a local maximum at x = c, the property f(c) = f(-c) implies that the local maximum is mirrored at x = -c, meaning f(-c) is also a local maximum.

Transcript

00:01 So these are some true and false and multiple choice questions.

00:03 I'm just going to do them all in one video.

00:05 Some of them are short.

00:06 Some of them are longer.

00:08 So they ask us, if f double prime at c is zero, then c comma f of c is a point of inflection.

00:15 Justify your answer.

00:16 Well, that is false.

00:19 Because we have to have concavity changing at that point.

00:28 And graph has to as a tangent, but again, if the second derivative exists, then it should be that should be okay.

00:36 But the cancadity doesn't necessarily change when f double prime is zero.

00:41 All we know that it's the first derivative is flat there.

00:44 So it's it's it's a very flat function at that point.

00:48 But you can take for example, you know f of x equals x to the fourth well f double prime is clearly zero at when x equals zero but you know second derivative looks like x squared so that's going to be the same sign.

01:05 Well, that's going to be, you know, there's a positive function out here, right? 12 out here, 12 x squared.

01:12 So that's always going to be positive.

01:15 So we know that the concavity doesn't change for this, even though the first second derivative is zero at zero.

01:23 So that's, this is not, it's not, the statement is not true.

01:29 Now the second one, they say if f prime is at c is zero and f double prime, that c is less than zero, then fc is a local maximum.

01:38 Well that that is true and you can go look in the chapter by the second derivative test, but you know, that means if f fc prime at c is zero, then we have a horizontal tangent if f double prime at c is negative then the slope must change sign and function must be below.

01:57 So we have to have something that looks like this locally.

02:01 So that means that c is the maximum.

02:06 Now, let's see here.

02:08 They ask us to, look at this.

02:12 So we have a cubic with an unknown first coefficient.

02:20 And they ask us about the, when is it concave up? so we take a first derivative and a second derivative.

02:27 The second of the derivative is 6ax plus 6.

02:32 So if this is greater than zero, then we have concave up.

02:38 And that is true on the range from minus, let's see here.

02:45 If it's concave up, let's see here.

02:52 You need it to be greater than zero...