## Maxima, minima and zeros

Maxima, minima and zeros of quadratic functions Multiplication of Decimals

#### ZEROS

Let f(x) = ax2+bx+c represent a quadratic function.

If f(x) = 0

= ax2+bx+c = 0

Then, the roots of the quadratic equation are also called the zeros of the quadratic function.

#### MAXIMA

If f(x)=ax2+bx+c gradually increases until it reaches a value p, which is algebraically greater than its neighboring values on either side, then p is said to be the maximum value or maxima of f(x).

#### MINIMA

If f(x)= ax2+bx+c gradually increases until it reaches a value q, which is algebraically less than its neighboring values on either side, q is said to be the minimum value or minima of f(x).

These concepts can be visualized by drawing graphs.

Example : 1

Draw the graph for x2+x-6 and find its zeros, maxima and minima, if they exist.

y = x2+x-6

 x -4 -3 -2 -1 0 1 2 3 x2 16 9 4 1 0 1 4 9 x -4 -3 -2 -1 0 1 2 3 -6 -6 -6 -6 -6 -6 -6 -6 -6 y -6 0 -4 -6 -6 -4 0 6

Plot the points (-4,6), (-3,0), (-2,-4), (-1,-6), (0,-6),(1,-4),(2,0),(3,6) and join them to form a smooth curve.

From the graph, the curve cuts the x-axis at (-3,0) and (2, 0).

So the zeros of x2+x-6 = 0 are x = -3,2.

The maxima does not exist, as the curve continues indefinitely on both sides.

The minima is obtained by drawing a line from point L to M on the x-axis.

LM=-6.25, which is the minimum value or minima and it occurs at x=-0.5. #### Try the following question

Draw the graph for the function f(x)= 4-5x-x2. From the graph obtain the zeros, maxima and minima, if they exist.

Let y = 4-5x-x2

 x -6 -5 -4 -3 -2 -1 0 1 2 -x2 -36 -25 -16 -9 -4 -1 0 -1 -4 -5x 30 25 20 15 10 5 0 -5 -10 4 4 4 4 4 4 4 4 4 4 y -2 4 8 10 10 8 4 -2 -10

Plot the points (-6,-2), (-5,4),(-4,8),(-3,10),(-2,10),(-1,8), (0,4),(1,-2),(2,-10) and join them to form a smooth curve.

From the graph, we see that the curve cuts the x-axis at the points (-5.7,0) and (0.7,0). So the zeros of 4-5x-x2 = 0 are x = -5.7, 0.7

Let L be the point where the curve is maximum. Drop a perpendicular from L to the x-axis, meeting the x-axis at M.

LM = 10.25. This is the maximum value and it occurs at

x = -2.5.

There is no minimum value as the curve continues on indefinitely on both sides. 