Functions and Finding the Extreme Value

Question

Given the function f(x)= 23−2(x+3)^2 find all of the following: 

1) Find the extreme value of the function:

2) The smallest root is:

3) The largest root is:

Answer 1

desmos.com/calculator/kojnihfzbn

Answer 2

it's a downward opening parabola with vertex (-3,21) = the maximum = the absolute and relative maximum

there is no absolute or relative minimum, as the function approaches negative infinity

23=2(x+3)^2 = 0 solve for x to find the two "roots," "zeros" "solutions or x intercepts, the two points where the curve crosses the x axis

2(x^2 +6x +9) = 23

2x^2 +12x +18-23=0

2x^2 +12x -5 = 0

use the quadratic formula

x =-12/4 +/-.25sqr(144+40)

x=-3 + or - (sqr184)/4

larger root = -3+(1/4)sqr184= -3+(1/2)sqr46= -3+sqr11.5= about 0.4

smaller root = -3-(1/4)sqr184= -3-(1/2)sqr46=-3-sqr11.5 = about -6.4

or complete the square

2x^2 +12x =5

x^2 +6x = 5/2

x^2+6x +9 = 2.5+9 = 11.5

x+3 = +/-sqr11.5

x =-3 +/-sqr11.5= about 0.4 and -6.4

notice the two roots are equidistant from the x coordinate of the vertex, 3

3+/- 3.4= .4, -6.4