`f(x)=3x^2+6x+7` Given that f(x) can be written in the form `A(x+B)^2+C` where A, B and C are rational numbers. Find the value of A, B and C Hence/Otherwise, find: a) the value of x for which`f(x)` is a minimum b) The minimum value of `f(x)` Step by step answers
- print Print
- list Cite
Expert Answers
calendarEducator since 2012
write251 answers
starTop subjects are Math and Literature
1) The number beside `x^2` is A, so:
`f(x)=3(x+B)^2+C`
2) Factor out A. That is, divide everything by 3:
`f(x)=3(x^2+2x+7/3)`
3) The number that is next to `x` is double B
So, to find B, take 2 and cut in half
So B =1
`f(x)=3(x+1)^2+C`
4) There are a few ways to find C. Here is one of them:
You want
``
``
``
``
``
`4=C`
So:
`f(x)=3(x+1)^2+4`
One of the reasons this is useful, is it immediately tells you the minimum of the parabola. (If A is negative, it is an upside-down parabola, and instead the parabola has a maximum)
The minimum value occurs at `x=-B`
So, for this problem, the minimum occurs at `x=-1`
Plug in to find what the minimum value is:
`f(-1)=3(-1+1)^2+4=0+4=4`
That is, the minimum value is just C
To summarize:
If `f(x)=A(x+B)^2+C`
then the minimum occurs at `x=-B` , and the minimum is `C`
(if A is negative, then the maximum occurs at -B, and the maximum is C)
So, for us, the value of `x` for which `f(x)` is a minimum is -1
and the minimum value of `f(x)` is 4
Related Questions
- `f(x) = (x^2) - x - ln(x)` (a) Find the intervals on which `f` is increasing or decreasing....
- 2 Educator Answers
- `f(x) = x/(x^2 + 1)` (a) Find the intervals on which `f` is increasing or decreasing. (b)...
- 1 Educator Answer
- Suppose that 2 ≤ f '(x) ≤ 5 for all values of x. What are the minimum and maximum possible values...
- 1 Educator Answer
- `f(x) = x^4 - 2x^2 + 3` (a) Find the intervals on which `f` is increasing or decreasing. (b)...
- 1 Educator Answer
- `f(x) = (x^2 - 1)^3, [-1, 2]` Find the absolute maximum and minimum values of f on the...
- 1 Educator Answer