Extrema of Quadratic functions

In summary, it seems as if all quadratic functions would have a relative extrema, but there may be exceptions depending on the form of the derivative.
  • #1
Qube
Gold Member
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1

Homework Statement



Does every quadratic function have a relative extrema?

Homework Equations



Quadratic function: ax^2 + bx + c. Aka a polynomial.

Polynomials are continuous through all real numbers.

The Attempt at a Solution



It seems as if all quadratic functions would have a relative extrema since the basic shape of a quadratic function is U-shaped and it's graphically obvious that the second derivative changes sign; all quadratic functions have a vertex whose x coordinate is given by -b/2a and this vertex is also the location of a horizontal tangent line and always represents either the max or min of the quadratic function (depending on orientation). And taking the derivative of the general form of a quadratic function yields 2ax + b where a and b are constants and it would appear that one can easily make the derivative both positive and negative given that the domain of quadratic functions is all real numbers.

Are there any exceptions? (I'm guessing no).
 
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  • #2
Qube said:
It seems as if all quadratic functions would have a relative extrema since the basic shape of a quadratic function is U-shaped and it's graphically obvious that the second derivative changes sign; all quadratic functions have a vertex whose x coordinate is given by -b/2a and this vertex is also the location of a horizontal tangent line and always represents either the max or min of the quadratic function (depending on orientation). And taking the derivative of the general form of a quadratic function yields 2ax + b where a and b are constants and it would appear that one can easily make the derivative both positive and negative given that the domain of quadratic functions is all real numbers.
Right

Are there any exceptions? (I'm guessing no).
No. Just make sure that "quadratic functions" implies a!=0 in your formula (otherwise it is not a quadratic function).
 
  • #3
Qube said:

Homework Statement



Does every quadratic function have a relative extrema?

Homework Equations



Quadratic function: ax^2 + bx + c. Aka a polynomial.

Polynomials are continuous through all real numbers.

The Attempt at a Solution



It seems as if all quadratic functions would have a relative extrema since the basic shape of a quadratic function is U-shaped and it's graphically obvious that the second derivative changes sign
The second derivative is constant. You mean the first derivative changes sign.

By the way, extrema is the plural of extremum. You should say that quadratic functions have a relative extremum.
 
  • #4
Right, and looking at the first derivative, it can only be positive since it's the product of three squared terms. So there can be no sign change regardless.
 
  • #5
Qube said:
Right, and looking at the first derivative, it can only be positive since it's the product of three squared terms. So there can be no sign change regardless.
I guess this belongs to your other thread.
 
  • #6
Qube said:
Right, and looking at the first derivative, it can only be positive since it's the product of three squared terms. So there can be no sign change regardless.
?
"it" = what?
If y = ax2 + bx + c, then y' = 2ax + b

Where are you getting the three squared terms? mfb seems to know, but I don't recall seeing that other thread.

If "it" refers to y', the derivative will always change sign in a quadratic function.

If "it" refers to y'', that's 2a, so I still don't see where the three squared terms business comes in.
 
  • #7
mfb said:
I guess this belongs to your other thread.

Whoa LOL yes this belongs in the other thread. Confused. My apologies.
 
  • #8

1. What is an extrema of a quadratic function?

The extrema of a quadratic function is the point where the function reaches its maximum or minimum value. This point can be found by finding the vertex of the parabola that represents the quadratic function.

2. How do you find the extrema of a quadratic function?

The extrema of a quadratic function can be found by using the formula x = -b/2a, where a and b are the coefficients of the quadratic function in the form of ax^2 + bx + c. This x value represents the x-coordinate of the vertex, and the y-coordinate can be found by plugging in this x value into the original function.

3. Can a quadratic function have more than one extrema?

Yes, a quadratic function can have two extrema, a maximum and a minimum. However, it is also possible for a quadratic function to have only one extrema, depending on the shape and position of the parabola.

4. How do you determine if the extrema of a quadratic function is a maximum or a minimum?

The extrema of a quadratic function is a maximum if the coefficient of x^2 is negative, and a minimum if the coefficient is positive. This can also be determined by looking at the concavity of the parabola, where a downward concave parabola represents a maximum and an upward concave parabola represents a minimum.

5. Can the extrema of a quadratic function be at the endpoints of the function's domain?

No, the extrema of a quadratic function can only occur at the vertex or at a point within the domain of the function. The endpoints of the function's domain can only be the maximum or minimum values if the quadratic function is a constant function, in which case the extrema would be at every point within the domain.

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