Let f(x)=x^4-3 x^2+1 (a) Show that f(x) is an even function....

Key Concepts

Polynomials and Symmetry

Polynomials, as algebraic expressions consisting of terms with nonnegative integer exponents, can be classified as even or odd based on the exponents of x in their terms. A polynomial is even if all its nonzero terms have even exponents, which implies symmetry about the y-axis. Recognizing the symmetry properties of polynomials is useful because it influences the behavior of their derivatives and integrals, and it allows for simplification in many algebraic and calculus problems.

Derivative and Symmetry

A key concept in differentiation is how it interacts with function symmetry. Specifically, when a function is even, its derivative is odd. This is because differentiating the even function f(x) = f(–x) with respect to x involves applying the chain rule, which introduces a sign change in the derivative, hence making it odd. This concept is fundamental in calculus and analysis when determining the nature of the derivative of symmetric functions.

Even Functions

An even function is one that satisfies f(x) = f(–x) for all x in its domain. This symmetry about the y-axis means that the function's value does not change when x is replaced with its negative. Such functions are important in many areas of mathematics because they often simplify integration, series representation and analysis of symmetry in graphs.

Odd Functions

An odd function satisfies f(–x) = –f(x) for all x in its domain, indicating symmetry about the origin. This property means that the function’s graph, when rotated 180° around the origin, remains unchanged. Understanding odd functions is crucial for symmetry considerations and simplifying problems in calculus, especially when integrating over symmetric intervals.

Transcript

00:01 So we're given this function f of x, and what we want to find is if this is an even function.

00:05 And so for a function to be even, we need f of negative x to be equal to f of x.

00:11 So whenever we plug in a negative x for our x, we should get the same function f of x out.

00:17 So let's see if that happens.

00:19 So if we plug in negative x into our equation, you have negative x to the fourth minus three negative x squared plus one.

00:29 And we know that negative 1 to an even power 2, 4 ,6, any even power is going to be equal to 1.

00:37 So negative x to the 4th is actually just equal to x to the 4th, and negative x squared is just equal to x squared.

00:44 So we have minus 3x squared plus 1.

00:47 And if we look at this, this is actually equal to our f of x.

00:51 So we do indeed have a even function.

00:55 So f of x is even.

01:02 And now for part b, what we want to find is if the derivative, f prime of x, is odd.

01:08 So let's first find our derivative, and we're just going to use the power rule to find this.

01:12 So we bring down the exponent, we minus 1, and we do the same thing for 3x squared.

01:17 So we are left with negative 6x.

01:21 And for a function to be odd, what we need is we need f prime of negative x to be equal to negative.

01:31 F prime of x and so we're just going to see if this happens when we plug it negative x so f prime of negative x is equal to four negative x to the third minus six times negative x so negative one to the third is still negative one so we have a negative four x to the third and six times negative x is negative six x and we have this minus so we have plus six x and now we can take a negative one out of this and we can say negative one times 4 x to the third minus 6x is equal to f prime of negative x and if we look at our original equation f prime of x this is actually equal to negative f prime of x so this is indeed a odd function since this is equal to negative f prime of x so f prime of x is odd.

02:30 And for part c, you want to see if any function g of x that is an even function has a derivative that is an odd function...

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