Answer to Question #103927 in Trigonometry for Martin

Question

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

Answer 1

Solution

Find the first derivative.

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

Equate the first derivative to zero and solve the equation.

$2x-2=0$

$x=1$

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.

$x=1,-4$

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

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

We have that:

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

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