Determine the location and values of the absolute maximumu and minimum for given function f(x)=(-x+2)^4 minimum

for a function f(x), the maximum is defined as:

f(x₀), such that:

f(x₀) ≥ f(x) for all values of x.

and a minimum is defined as:

f(xₐ), such that:

f(xₐ) ≤ f(x) for all values of x.

In this case, we have the function:

f(x) = (-x + 2)^4

The first thing you should see is that the exponent is even, then f(x) is always equal or larger than zero.

Then the smaller value that f(x) can take is zero, and this happens when the argument inside the parentheses is equal to zero:

-x + 2 = 0

2 = x

so the minimum is:

f(2) = (-2 + 2)^4 = 0

For any value different than x = 2, f(x) is larger than zero.

Now, we can also see that, as x increases, f(x) also does increase (if we graph the equations we could see that the graph extends infinitely upwards) then this function does not have an upper limit, thus, the function has not a maximum.

given that triangle abc is transformed two times to get triangle

a"b"c"

first abc is transformed in to a'b'c' as follows:

the coordinates of a are (1,-1) and transformed into a'(-1,1)

similarly b (4,-2) became b'(-4,2) and

c(7,-2) became c'(-7,2)

i.e. (x,y) becomes (-x,-y)

this is nothing but reflection about a point here origin.

thus first transformation is reflection on the origin.

next is exactly shifting 3 units down

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