Toán
tìm giá trị lớn nhất của hàm số y=-x+5-2√x-2

answer:

f(x) has a global maximum at x = 0

step-by-step explanation:

find and classify the global extrema of the following function:

f(x) = -x + 5 - 2 sqrt(x) - 2

find the critical points of f(x):

compute the critical points of 3 - 2 sqrt(x) - x

the domain of f(x) = 3 - 2 sqrt(x) - x is {x element r : x> =0}

the interior of the domain is {x element r : x> 0}:

f(x) = 3 - 2 sqrt(x) - x when x> 0

to find all critical points, first compute f'(x):

d/( dx)(3 - 2 sqrt(x) - x) = -1/sqrt(x) - 1:

f'(x) = -1/sqrt(x) - 1

f'(x) is never zero when x> 0:

-1/sqrt(x) - 1 is never zero when x> 0

f'(x) exists for all x such that x> 0:

-1/sqrt(x) - 1 exists for all x such that x> 0

f(x) has no critical points:

3 - 2 sqrt(x) - x has no critical points

the domain of 3 - 2 sqrt(x) - x is {x element r : x> =0}:

the endpoints of {x element r : x> =0} are x = 0 and ∞

evaluate 3 - 2 sqrt(x) - x at x = 0 and ∞:

the open endpoints of the domain are marked in gray

x | f(x)

0 | 3

∞ | -∞

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

0 | 3 | global max

∞ | -∞ | global min

remove the points x = ∞ from the table

these cannot be global extrema, as the value of f(x) here is never achieved:

x | f(x) | extrema type

0 | 3 | global max

f(x) = 3 - 2 sqrt(x) - x has one global maximum:

answer: f(x) has a global maximum at x = 0

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