1. The problem statement, all variables and given/known data
Does every quadratic function have a relative extrema?
2. Relevant equations
Quadratic function: ax^2 + bx + c. Aka a polynomial.
Polynomials are continuous through all real numbers.
3. 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).
Does every quadratic function have a relative extrema?
2. Relevant equations
Quadratic function: ax^2 + bx + c. Aka a polynomial.
Polynomials are continuous through all real numbers.
3. 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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