How to Solve Functional Inequality with Multiple Unknowns?

[Michael_Light]

Homework Statement

Given http://www.mathhelpforum.com/math-help/attachments/f33/20928d1298610998-function-msp281219ebge8he857gc6900005ba9285dff0f5h79.gif , find the values of ''a'' for which the value of the function f(x) <= 25/2.

The answer is a<= 1/2.

Homework Equations

The Attempt at a Solution

I have no ideas how to eliminate the x, i can't solve it cause there are 2 unknown in one inequality...

 

[tiny-tim]

Hi Michael!

(have an leq: ≤ and try using the X² icon just above the Reply box )

… find the values of ''a'' for which the value of the function f(x) <= 25/2.
…
I have no ideas how to eliminate the x, i can't solve it cause there are 2 unknown in one inequality...

It means find a such that the maximum value of f(x) is 25/2.

Hint: complete the square ​

 

[Michael_Light]

Hi Michael!

(have an leq: ≤ and try using the X² icon just above the Reply box )


It means find a such that the maximum value of f(x) is 25/2.

Hint: complete the square ​

How do you know that it is a maximum but not minimum? By letting the maximum of f(x) = 25/2, i got a=1/2, what should i do next?

 

[tiny-tim]

By letting the maximum of f(x) = 25/2, i got a=1/2, what should i do next?

That sounds like an answer.

How did you get it?

Doesn't the way you got it tell you whether it's a maximum or minimum?

 

[Michael_Light]

Hi Michael!

(have an leq: ≤ and try using the X² icon just above the Reply box )


It means find a such that the maximum value of f(x) is 25/2.

Hint: complete the square ​

I mean... how do you know that it is a maximum value before you find ''a''? and does f(x) <= 25/2
indicates that the maximum/minimum value of f(x) is smaller or equal than 25/2? Thanks.

 

[tiny-tim]

I've no idea what you've done.

… i got a=1/2 …

how did you get a = 1/2 ?

 

[Michael_Light]

I've no idea what you've done. how did you get a = 1/2 ?

Max/min value of f(x), i.e -b²/4a + c = 25/2 and solve it...

 

[tiny-tim]

Max/min value of f(x), i.e -b²/4a + c = 25/2 ...

Is this just a formula that you've learned from somewhere, or do you know how to prove it?

 

[Michael_Light]

Is this just a formula that you've learned from somewhere, or do you know how to prove it?

By solving

.. i managed to get a=1/2, but yet the answer is a<= 1/2... i don't know why a <= 1/2...

 

[tiny-tim]

Do you know how to prove this formula??

Where did you get it from? ​

 

[Michael_Light]

Do you know how to prove this formula??

Where did you get it from? ​

From http://www.mathhelpforum.com/math-help/attachments/f33/20928d1298610998-function-msp281219ebge8he857gc6900005ba9285dff0f5h79.gif , we get f(x)=(a-1)x²+(4a+3)x+(4a-2)... where A represents (a-1), B represents (4a+3) and C represents (4a-2)... so -B²/4A + C = maximum/minimum value of f(x)...

 

[tiny-tim]

From … we get f(x)=(a-1)x²+(4a+3)x+(4a-2)... where A represents (a-1), B represents (4a+3) and C represents (4a-2)... so -B²/4A + C = maximum/minimum value of f(x)...

Yes, I know you can apply the formula (-B²/4A + C), but can you prove it?

Where did you get it from?​

 

[Michael_Light]

Yes, I know you can apply the formula (-B²/4A + C), but can you prove it?

Where did you get it from?​

From my reference book.

 

[tiny-tim]

ok then!

take the equation ax² + bx + c = 0 and complete the square …

what do you get? ​

 

[Michael_Light]

ok then!

take the equation ax² + bx + c = 0 and complete the square …

what do you get? ​

Someone clarified it for me... I managed to solve it now... Thanks for your time and patient..