f(x)=6x^2−10x+10,0≤x≤9

Question

What is the absolute maximum and minimum of this question on the given interval

Answer 1

f(x)=6x^2−10x+10,0≤x≤9

What is the absolute maximum and minimum of this function on the given interval.

y=6x²-10x+10 is a concave up parabola.

dy/dx=12x-10=0

d²y/dx²=12 > 0 so that extrema is a minimum.

and x=10/12=5/6 at the vertex

when x=5/6, y=6(5/6)²-10(5/6)+10=25/6-50/6+10=-25/6+60/6=35/6≅5.83

so the vertex (minimum) is at (5/6,35/6)

The interval is [0,9]

When x=0, y=10

When x=9, y=6(9)²-10(9)+10=6(81)-90+10=406

Absolute maximum is 406

Absolute minimum is 35/6≅5.83

Answer 2

f'(x) = 12x - 10

0 = 12x - 10

10/12 = x

x = 5/6

f(5/6) = 6 ( 5/6)^2 - 10( 5/6) + 10

= 25/6 - 50/6 + 60/6

= 35/6 = 5 and 5/6

f(0) = 10

f(9) = 6(9)^2 - 10(9) + 10 =

6(81) - 90 + 10 =

486 - 90 + 10 =

406

absolute max is 406

absolute min is 5 and 5/6

Answer 3

The minimum is 10 and the maximum is f(9) which I will leave to you to compute.

The easiest way to do this problem is to graph it.