# 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

$f(x) = ax^2 + bx + c$

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

The first derivative of $$f$$ is given by

$f'(x) = 2ax + b$

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

$2ax + b > 0$ which may be written as $2 a x > - b$

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

$x > -\frac{b}{2a}$

We now use a table to analyze the sign of $$f'$$ 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 $$(- \infty, -b/2a)$$ and increasing over the interval $$(-b/2a, + \infty)$$.

### case 2: coefficient $$a \lt 0$$

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

$x \lt -\frac{b}{2a}$

We now analyze the sign of $$f'$$ using the table below

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

## B - Quadratic Function in Vertex form

Quadratic functions in their vertex form are written as

$f(x) = a(x - h)^2 + k$

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

The first derivative of $$f$$ is given by

$f'(x) = 2a(x - h)$

We analyze the sign of $$f'$$ using a table. $$f'(x)$$ is positive if

$a(x - h) > 0$

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

$x > h$

The table below is used to analyze the sign of $$f'$$.

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

### case 2: coefficient $$a \lt 0$$

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

$x \lt h$

We analyze the sign of $$f'$$ using the table below

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

### Example 1

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

$f(x) = 2x^2 - 8x + 1$

### Solution to Example 1

• We first find the derivative
$$f'(x) = 4x - 8$$
$$f'(x)$$ 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 $$(- \infty, 2)$$ and increasing on $$(2, + \infty)$$. 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

$f(x) = - (x + 3)^2 + 1$

### Solution to Example 2

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

### Exercises on Properties of Quadratic Functions

For each quadratic function below find the extremum (minimum or maximum), the interval of increase and the interval of decrease.

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) $$f(x) = -x^2$$

a) minimum at the point $$(-3, -9)$$
decreasing on $$(-\infty, -3)$$
increasing on $$(-3, +\infty)$$

b) maximum at at the point $$(-1, 4)$$
increasing on the interval $$(-\infty, -1)$$
decreasing on the interval $$(-1, +\infty)$$

c) minimum at at the point $$(0 , -5)$$
decreasing on the interval$$(-\infty, 0)$$
increasing on the interval $$(0, +\infty)$$

d) maximum at $$(4, 2)$$
increasing on the interval $$(-\infty, 4)$$
decreasing on the interval $$(4, +\infty)$$

e) maximum at $$(0, 0)$$
increasing on the interval $$(-\infty, 0)$$
decreasing on the interval $$(0, +\infty)$$

### More on applications of differentiation

applications of differentiation