minimizing average value

[serenamae1216]

The question I am given is Find the value of a that minimizes the average value of f(x)=x⁴-2x² on the interval [a,a+2]

I started out by using the formula for the average value of a function to find that (1/(a+2)-a)∫x⁴-2x²dx with the limits of integration being a to a+2 is equal to the average value of the function. I then change the fraction at the beginning to just one of two since the two a's will cancel. rom this i tried to integrate and came to (½)[(1/5(a+2)⁵+⅔(a+2)³)-(1/5a-2/3a³) This is where I got stuck. Can anyone give me some help on what I should do next?

 

[stapel]

You're trying to "minimize", right? So what process often comes to play in a max/min situation?

 

[Deleted member 4993]

The question I am given is Find the value of a that minimizes the average value of f(x)=x⁴-2x² on the interval [a,a+2]

I started out by using the formula for the average value of a function to find that (1/((a+2)-a))∫(x⁴-2x²)dx with the limits of integration being a to a+2 is equal to the average value of the function. I then change the fraction at the beginning to just one of two since the two a's will cancel. rom this i tried to integrate and came to (½)[(1/5(a+2)⁵- ⅔(a+2)³)-(1/5a⁵-2/3a³) This is where I got stuck. Can anyone give me some help on what I should do next?

In other words - what Stapel said -

Now you have a function of 'a' which is f(a) - and you need to find the the value of 'a' where f(a) is minimum.

You had several mistakes in your integration - watch those.

I would also suggest that you should plot x^4-2x^2 and investigate the domain for minimum average value.