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Proof, EXT2 P1 2022 HSC 16c

It is given that for positive numbers `x_(1),x_(2),x_(3),dots,x_(n)` with arithmetic mean `A`,
 

               `(x_(1)xxx_(2)xxx_(3)xx cdots xxx_(n))/(A^(n)) <= 1`    (Do NOT prove this.)

Suppose a rectangular prism has dimensions  `a,b,c`  and surface area `S`.

  1. Show that  `abc <= ((S)/(6))^((3)/(2))`.  (2 marks)

  2. Using part (i), show that when the rectangular prism with surface area `S` is a cube, it has maximum volume.  (2 marks)

Show Answers Only
  1. `text{Proof (See Worked Solutions)}`
  2. `text{Proof (See Worked Solutions)}`
Show Worked Solution

i.    `text{Show}\ \ abc <= ((S)/(6))^((3)/(2))`

`S=2(ab+bc+ac)`

`A=(ab+bc+ac)/3`

`text{Using given relationship, where}\ x_1=ab, x_2=bc, …`


♦♦♦ Mean mark (i) 26%.
`(ab xx bc xx ca)/((ab+bc+ac)/3)^3` `<=1`  
`(ab xx bc xx ca)` `<=((ab+bc+ac)/3)^3`  
`(abc)^2` `<=((2(ab+bc+ac))/6)^3`  
`abc` `<=(S/6)^(3/2)`  

 

ii.   `text{If prism is a cube}\ \ a=b=c`

`=> V=a^3, \ \ S=6a^2`

`(S/6)^(3/2)=((6a^2)/6)^(3/2)=a^3`
 

`text{In the case of a cube:}`

`V=(S/6)^(3/2)`


♦♦♦ Mean mark (ii) 26%.

`text{Also,}\ \ V<=(S/6)^(3/2)\ \ \ text{(using part (i))}`

`:.\ text{For rectangular prisms with a given surface area, a cube}`

`text{has the maximum volume.}`

Proof, EXT2 P1 2017 HSC 10 MC

Suppose  `f(x)`  is a differentiable function such that

`(f(a) + f(b))/2 >= f((a + b)/2)`, for all `a` and `b`.

Which statement is always true?

  1. `int_0^1 f(x)\ dx >= (f(0) + f(1))/2`
  2. `int_0^1 f(x)\ dx <= (f(0) + f(1))/2`
  3. `f′(1/2) >= 0`
  4. `f′(1/2) <= 0`
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`B`

Show Worked Solution

`text(Representing the equation graphically)`

♦ Mean mark 41%.

`text{(one of many possibilities)}`

 

 

`f(x)\ text(is an increasing function between)`

`a\ text(and)\ b\ text(such that:)`

`(f(a) + f(b))/2 >= f((a + b)/2)`
 

`text(If)\ \ a = 0\ text(and)\ \ b = 1,`

`int_0^1 f(x)\ dx` `<=\ text(Area of trapezium)`
  `<= 1/2(1) (f(0) + f(1))`
  `<= (f(0) + f(1))/2`

`=> B`