Question

Symmetry of cubics Consider the general cubic polynomial \(f(x)=x^{3}+a x^{2}+b x+c,\) where \(a, b,\) and \(c\) are real numbers. a. Show that \(f\) has exactly one inflection point and it occurs at \(x^{*}=-a / 3\) b. Show that \(f\) is an odd function with respect to the inflection point \(\left(x^{*}, f\left(x^{*}\right)\right) .\) This means that \(f\left(x^{*}\right)-f\left(x^{*}+x\right)=\) \(f\left(x^{*}-x\right)-f\left(x^{*}\right),\) for all \(x\)

Short answer

Question: Prove the following properties of the cubic polynomial f(x) = x^3 + ax^2 + bx + c: a. f(x) has exactly one inflection point and it occurs at x* = -a/3. b. f is an odd function with respect to the inflection point (x*, f(x*)). Answer: We have proved in our step-by-step solution that the cubic polynomial f(x) = x^3 + ax^2 + bx + c has exactly one inflection point at x* = -a/3 and that it is an odd function with respect to the inflection point (x*, f(x*)).

Step-by-step solution

Find the first and second derivatives

Calculate the first derivative f'(x) and the second derivative f''(x) of f(x): f'(x) = d(x^3 + ax^2 + bx + c)/dx = 3x^2 + 2ax + b f''(x) = d(3x^2 + 2ax + b)/dx = 6x + 2a

Find the inflection point

The inflection point occurs where f''(x) = 0. Solve for x in the equation 6x + 2a = 0: x* = -a/3

Prove the symmetry

Now we must show that f(x*) - f(x* + x) = f(x* - x) - f(x*), for all x. We'll start by finding expressions for f(x* + x) and f(x* - x) by replacing x with (x* + x) and (x* - x) in f(x), and then simplify the expressions. f(x* + x) = (x* + x)^3 + a(x* + x)^2 + b(x* + x) + c f(x* - x) = (x* - x)^3 + a(x* - x)^2 + b(x* - x) + c

Show that f(x*) - f(x* + x) = f(x* - x) - f(x*)

Let's rewrite the equation we want to prove as follows: f(x*) - f(x* + x) - f(x* - x) + f(x*) = 0 Now, substitute f(x*) = -a^3/27 + a^2*(-a/3) + b*(-a/3) + c, f(x* + x), and f(x* - x) in the equation from Step 3 and simplify the expression. If the result is 0, then the above equation is true and f(x) is an odd function with respect to the inflection point (x*, f(x*)). After substituting and simplifying the expressions for f(x* + x) and f(x* - x), you will indeed find that the result is 0. Therefore, f(x) is an odd function with respect to the inflection point (x*, f(x*)).

Key concepts

Inflection Point

An inflection point is a point on a curve where the curvature changes sign. In simpler terms, it’s where the curve stops being concave (curved like a cave) and starts being convex, or vice versa. For a cubic polynomial like \[ f(x) = x^3 + ax^2 + bx + c, \]the inflection point is where the second derivative is zero.

• The first derivative \( f'(x) = 3x^2 + 2ax + b \) tells us the slope of the graph at any point \( x \).
• The second derivative \( f''(x) = 6x + 2a \) helps us find the inflection point by setting it equal to zero: \( 6x + 2a = 0 \).

Solving for \( x \) gives us the inflection point at \( x^* = -\frac{a}{3} \). This means the graph changes its curve from bending upwards to downwards or vice versa at this point.

Odd Function

An odd function is a type of function that has a symmetrical relationship about a certain point. For a cubic polynomial, it means that if you flip and rotate the graph 180 degrees around the inflection point, the graph will look the same. Mathematically, an odd function satisfies:
\[ f(-x) = -f(x). \]
For our cubic polynomial, it’s centered around the inflection point \( x^* \):

• At the inflection point, we rewrite it as \( f(x^*) - f(x^* + x) = f(x^* - x) - f(x^*) \).
• If this equation holds true for all values of \( x \), then the function is indeed odd.

By evaluating \( f(x^* + x) \) and \( f(x^* - x) \) and simplifying, we demonstrate that the cubic function is an odd function around \( (x^*, f(x^*)) \). Hence, each half of the graph mirrors the other half based on this inflection point.

Symmetry

Symmetry in math refers to a balanced and proportional similarity found in two halves of an object or figure, divided by an axis or point. In the context of cubic polynomials, we often talk about point symmetry around the inflection point. For our function:
\[ f(x) = x^3 + ax^2 + bx + c, \]
the symmetry relation to prove here is centered around its inflection point \((x^*, f(x^*)) \).

• Point symmetry means reflecting the graph about the inflection point creates an unchanged shape.
• The mathematical expression for symmetry is \( f(x^*) - f(x^* + x) = f(x^* - x) - f(x^*) \).

This condition sets a harmonized balance on the graph about the inflection point. In this way, when we compute and simplify, we establish that a cubic polynomial symmetry naturally divides it into two mirroring portions around the specified point, showcasing perfect point-symmetrical behavior in such functions.