Question

Determine the location and absolute maximum and absolute minimum for the given function f(x)=(_x+2)^4 where 0

Answer 1

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.