Answer:
the minimum is 0 and the maximum is 16.
Step-by-step explanation:
For a function f(x), the maximum is defined as:
f(xₐ)
such that:
f(xₐ) ≥ f(x) for every value of x in the domain.
And the minimum is:
f(xₙ)
Such that:
f(xₙ) ≤ f(x)
for every value of x.
Now, in this case we have the function:
f(x) = (-x + 2)^4 in the range 0 < x < 3
First, let's analyze our function.
We can see that the exponent is even, then:
f(x) = (-x + 2)^4
Can be equal to zero or larger than zero.
Then the minimum will be f(x) = 0
To find this, we need to find the value of x such that:
-x +2 = 0
2 = x
Then the minimum is:
f(2) = (-2 + 2)^4 = 0
For the maximum, we can just play with the other values in the range and see which one gives the larger value of f(x).
Again, because f(x) = (-x + 2)^4 , the maximum will be at the value of x such that:
|-x + 2| is maximized.
As we have a negative sign multiplying x, the smaller value in the range:
x = 0
is the one that maximizes that:
|-0 + 2| = |2| = 2
Evaluating f(x) in x = 0 we get:
f(0) = (0 + 2)^4 = 16
Then the minimum is 0 and the maximum is 16.
