Quick min/max question: finding global max on interval

[jwpaine]

Lets say I have the function: f(x) = x^5 - x^10 and want to find the global max on the interval 0 <= x <= 1

if I set the first derivative of f(x) = 0:

5x^4 - 10x^9 = 0

x = .870551 or x = 0

I know how to test to see if it's a min or max by taking the second derivative, plugging in a point and seeing if it's less than or greater than 0 (or = to 0 for inflection)

Now my professor always says to check end points and always plugs in both ends of an interval into f(x) before checking for critical points.... why?

 

[Deleted member 4993]

Re: Quick min/max question

Lets say I have the function: f(x) = x^5 - x^10 and want to find the global max on the interval 0 <= x <= 1

if I set the first derivative of f(x) = 0:

5x^4 - 10x^9 = 0

x = .870551 or x = 0

I know how to test to see if it's a min or max by taking the second derivative, plugging in a point and seeing if it's less than or greater than 0 (or = to 0 for inflection)

Now my professor always says to check end points and always plugs in both ends of an interval into f(x) before checking for critical points.... why?

Because you are looking for Global min/max.

You may not have to check the end-points before checking the interior points - but you must check those before you decide on an answer.

 

[jwpaine]

Not to do with my problem, but question:

for a <= x <= b, what could f(a) or f(b) potentially show/give me?

 

[Deleted member 4993]

But for a <= x <= b, what could f(a) or f(b) potentially show/give me?

If the roots of f'(x) were x_i

It is possible that f(a) or f(b) > f(x_i)

In that case the global maxima is either f(a) or f(b) - not any of the f(x_i)s.

Converse will be true for global minima.