Answer to Question #103927 in Trigonometry for Martin

Question #103927
Find the absolute maximum and minimum values of f(x)=x−2x+4, if any, over the interval (−4,1].

absolute maximum is
and it occurs at x =

absolute minimum is
and it occurs at x
1
Expert's answer
2020-03-04T17:12:18-0500

Solution

Find the first derivative.

f(x)=2x2f'(x) =2x-2

Equate the first derivative to zero and solve the equation.

2x2=02x-2=0

x=1x=1

The roots of the equation are critical points. Use the end points of the interval and all critical points on it to check for an absolute extremum in this interval.

x=1,4x=1,-4

f(1)=1221+4=3f(1)=1^2-2*1+4=3

f(4)=(4)2(4)2+4=16+8+4=28f(-4)=(-4)^2-(-4)*2+4=16+8+4=28

We have that:

f(1)<f(4)f(1)<f(-4)

It means that absolute maximum could be f(x) =28, but x=-4 is out of the domain (hence it is not attained), the absolute minimum is f(x) =3 and it occurs at x=1.


Answer:

the absolute minimum is f(x)=3f(x) = 3 and it occurs at x=1, the absolute maximum is not reachable.


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