Derivative, Maximum, Minimum of Quadratic Functions

Differentiation is used to analyze the properties such as intervals of increase, decrease, local maximum, local minimum of quadratic functions.

A - Quadratic Function in General form

Quadratic functions in their general form are written as

where $a$, $b$, and $c$ are real numbers such that $a\ne 0$.

The first derivative of $f$ is given by

Let us analyze the sign of $f^{\prime }$ and hence determine any maximum or minimum point and the intervals of increase and decrease. $f^{\prime }(x)$ is positive if

We now need to consider two cases and continue solving the inequality above.

case 1: coefficient $a>0$

We divide both sides of the inequality by $2a$ and solve it to obtain

We now use a table to analyze the sign of $f^{\prime }$ and whether $f$ is increasing over a given interval.

The quadratic function with $a>0$ has a minimum point at $(-b/2a,f(-b/2a))$ and the function is decreasing on the interval $(-\mathrm{\infty },-b/2a)$ and increasing over the interval $(-b/2a,+\mathrm{\infty })$.

case 2: coefficient $a<0$

We divide both sides of the inequality by $2a$ but because $a$ is less than 0, we need to change the symbol of inequality

We now analyze the sign of $f^{\prime }$ using the table below

The quadratic function with $a<0$ has a maximum point at $(-b/2a,f(-b/2a))$ and the function is increasing on the interval $(-\mathrm{\infty },-b/2a)$ and decreasing over the interval $(-b/2a,+\mathrm{\infty })$.

B - Quadratic Function in Vertex form

Quadratic functions in their vertex form are written as

where $a$, $h$, and $k$ are real numbers with $a\ne 0$.

The first derivative of $f$ is given by

We analyze the sign of $f^{\prime }$ using a table. $f^{\prime }(x)$ is positive if

We need to consider two cases again and continue solving the inequality above.

case 1: coefficient $a>0$

We divide both sides of the inequality by $a$ and solve the inequality

The table below is used to analyze the sign of $f^{\prime }$.

The quadratic function with $a>0$ has a minimum at the point $(h,k)$ and it is decreasing on the interval $(-\mathrm{\infty },h)$ and increasing over the interval $(h,+\mathrm{\infty })$.

case 2: coefficient $a<0$

We divide both sides of the inequality by $a$ but we need to change the symbol of inequality because $a$ is less than 0.

We analyze the sign of $f^{\prime }$ using the table below

The quadratic function with $a<0$ has a maximum point at $(h,k)$ and the function is increasing on the interval $(-\mathrm{\infty },h)$ and decreasing over the interval $(h,+\mathrm{\infty })$.

Example 1

Find the extremum (minimum or maximum) of the quadratic function $f$ given by

Solution to Example 1

• We first find the derivative
  $f^{\prime }(x)=4x-8$
  $f^{\prime }(x)$ changes sign at $x=\frac{8}{4}=2$. The leading coefficient $a$ is positive hence $f$ has a minimum at $(2,f(2))=(2,-7)$ and $f$ is decreasing on $(-\mathrm{\infty },2)$ and increasing on $(2,+\mathrm{\infty })$. See graph below to confirm the result obtained by calculations.
  

Example 2

Find the extremum (minimum or maximum) of the quadratic function $f$ given by

Solution to Example 2

• The derivative is given by
  $f^{\prime }(x)=-2(x+3)$
  $f^{\prime }(x)$ changes sign at $x=-3$. The leading coefficient $a$ is negative hence $f$ has a maximum at $(-3,1)$ and $f$ is increasing on $(-\mathrm{\infty },-3)$ and decreasing on $(-3,+\mathrm{\infty })$. See graph below of $f$ below.
  

Exercises on Properties of Quadratic Functions

a) $f(x)=x^2+6x$
b) $f(x)=-x^2-2x+3$
c) $f(x)=x^2-5$
d) $f(x)=-(x-4{)}^2+2$
e) $f(x)=-x^2$

Answers to Above Exercises

More on applications of differentiation

applications of differentiation