What is the range of the function: h(x)=x+1/x^2 - 7x+10

  (-∞, -1-(2/3)√2] ∪ [-1+(2/3)√2, ∞)

Step-by-step explanation:

To make it easier to differentiate, we'll rewrite the function as ...

  h(x) = 2/(x-5) -1/(x-2)

Then the derivative is ...

  h'(x) = -2/(x-5)^2 +1/(x-2)^2

This will be zero when ...

  -2(x-2)^2 +(x-5)^2 = 0

  -2(x^2 -4x +4) +(x^2 -10x +25) = 0

  -x^2 -2x +17 = 0

  x^2 +2x +1 = 17 +1

  x +1 = ±√18 = ±3√2

  x = -1 ±3√2

The values of the function at these locations are ...

  h(-1-3√2) = -1 +(2/3)√2 ≈ -1.9428

  h(-1+3√2) = -1 -(2/3)√2 ≈ -0.0572

Then the range of h(x) is ...

  (-∞, -1-(2/3)√2] ∪ [-1+(2/3)√2, ∞)

a 4/6 = 9/12   not proportional

step-by-step explanation:

to find which are not proportional, we can use cross products

if it is true, they are proportional.   if it is not true, they are not proportional.

a 4/6 = 9/12

4* 12 = 6*9

48 = 54

not this is not proportional  

b 2/3 = 12/18  

2* 18 = 3 * 12

36 = 36

this is proportional

c 5/4 = 10/8  

5*8 = 4*10

40 = 40

this is proportional

d 7/5 = 35/25

7*25 = 5 * 35

175 = 175

is proportional