Finding Absolute Extrema on a Closed Interval

[silverbomb20]

Homework Statement

FIND THE ABSOLUTE MINIMUM AND ABSOLUTE MAXIMUM OF:

f(x) = 9x + 1/x
on the interval [1,3]

Homework Equations

The Attempt at a Solution

I don't know how to get started!

 

[lanedance]

hi silverbomb20 - any ideas or thoughts on what defines a maxima or minima?

 

[silverbomb20]

i know i need to use the power rule to get it started, but i don't know how to apply it to the fraction..


derivative of x ^n is n * x^(n-1) right?

i know the 9x would go to just 9
but i am unsure about the fraction

 

[Mark44]

Yes, d/dx(xⁿ) = nx^n-1. And 1/x = x^-1, so you can use the power rule on that.

 

[lanedance]

first part sounds good

so we're stuck on
$$\frac{d}{dx} (\frac{1}{x})$$

you could write it like below and use the power rule
$$\frac{d}{dx} (x^{-1})$$

or you could try and do it from first principles...

 

[silverbomb20]

okay so would it be 9 + -1x^-2 ?

im a history major having a lot of trouble with calculus...please help me

 

[lanedance]

yes, that looks correct

so when does

$$9-\frac{1}{x^2} = 0$$ ?

 

[silverbomb20]

in my mind...never.

f1(x) = 9 + -1x^-2

so i have to set it equal to zero right?

so 9 + -1x^-2=0

but that doesn't factor

 

[Mark44]

Multiply both sides by x^2.

Before we get (more) bogged down with minutiae, let's look at the strategy. Extreme values of a function f will happen at values of x for which f'(x) is zero, or at endpoints of the interval of definition.

 

[lanedance]

f'(x) represents a critical point which could be a maxima, minima or point of inflection, but you nee dto check to find which

as we are talking about absolute max & min we also need to check the boundaries of [1,3] ie the points x=1 and x=3

as for the local minima you're on your way
9 + -1x^-2=0

try first multiplying both sides of the equation by x^2

 

[silverbomb20]

okay so that's going to give me 9x^2=x^2?

or does that -1x^-2 not cancel?

 

[Mark44]

okay so that's going to give me 9x^2=x^2?

No. Try again.

or does that -1x^-2 not cancel?