Answer:
Explanation:
find and classify the global extrema of the following function
f(x)=(x-2)^2+5
compute the critical points of (x-2)^2+5
to find all critical points, first compute f(x)
f(x)=2(x-2)
solving 2(x-2)=0 yields x=2
x=2
f(x) exists everyhere
2(x-2) exists everyhere
the only critical point of (x-2)^2+5 is at x=2
x=2
the domain of (x-2)^2+ 5 is R
the endpints of R are x = -∞ and ∞
Evalute (x-2)^2+5 at x = -∞, 2 and ∞
the open endpoints of the domain are marked in gray
x () f(x)
-∞ ∞
2 5
∞ ∞
the largest value corresponds to a global maximum, and the smallest value corresponds to a global minimum:
the open endpoints of the domain are marked in gray
x () f(x) extrema type
-∞ ∞ global max
2 5 global min
∞ ∞ global max
remove the points x = -∞ and ∞ from the table
These cannot be global extrema, as the value of f(x) here is never achieved
x () f(x) () extrema type
2 5 global min
f(x) = (x-2)^2+5 has one global minimum
Answer:
f(x) has a global minimum at x = 2