Home > Precalculus > Local and Absolute Extrema

Over the x-value interval #[−10,10]#, what are the absolute extrema of #f(x)=x^2#?

Answer 1

The absolute maximum value is 100 at #x=\pm 10# and the absolute minimum value is 0 at #x=0#.

This is clear just by knowing the graph of #y=f(x)=x^2# as an upward-opening parabola with a vertex at the point #(0,0)#.

You can also use calculus (just a little preview if you are in precalculus). Since the derivative #f'(x)=2x#, the only critical point of the function is #x=0#. Now you compare the value of the continuous function #f# at the critical point and the endpoints #x=\pm 10# of the closed and bounded interval #[-10,10]#:

#f(0)=0^2=0#, #f(\pm 10)=10^2=100#.

This guarantees that the absolute maximum value is 100 at #x=\pm 10# and the absolute minimum value is 0 at #x=0#.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Sadie Ackerman

The absolute extrema of ( f(x) = x^2 ) over the interval ([-10, 10]) are as follows:

Absolute maximum: ( f(10) = 100 ) at ( x = 10 )

Absolute minimum: ( f(-10) = 100 ) at ( x = -10 )

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.