# Find Maxima Minima Of A Function

• Apr 8th 2008, 11:36 AM
rajdey1
Find Maxima Minima Of A Function
For the following function, find a point of maxima and a point of minima, if these exist

f(x) = 12x6 – 4x4 + 15x3 –1.
• Apr 8th 2008, 11:42 AM
topher0805
$f(x) = 12x^6-4x^4+15x^3-1$

First, find the derivative of the function:

$f'(x) = 72x^5-16x^3+45x^2$

Now find the critical points. These occur when the derivative function is equal to zero or when it is undefined. Since the function is never undefined, the critical points occur only when it is equal to 0.

So, find the critical points. Then, create a sign chart. Where the sign changes from positive to negative you have a maximum and where it changes from negative to positive you have a minimum.
• Apr 8th 2008, 11:46 AM
Moo
Hello,

Firstly, derivate the function.

$f'(x)=72x^5-16x^3+45x^2=x^2(72x^3-16x+45)$

0 annulates the derivative twice. But it's not a maxima or minima. Take a look at "inflexion point"

Now i'm afraid you will have to derivate $72x^3-16x$ to get its variations of sign.

Then, make a sign table of the derivative of f and find its maxima and minima (it's quite hard to write it down on a computer...)
• Apr 8th 2008, 12:11 PM
Jhevon
Quote:

Originally Posted by rajdey1
For the following function, find a point of maxima and a point of minima, if these exist

f(x) = 12x6 – 4x4 + 15x3 –1.

find f'(x) and f''(x)

critical points occur when f'(x) = 0 or is undefined. find all such points.

when you get the points, plug them into the second derivative f''(x)

if f''(x) < 0 at the point, then we have a maximum

if f''(x) > 0 at the point, then we have a minimum
• Apr 8th 2008, 12:12 PM
TheEmptySet
Quote:

Originally Posted by rajdey1
For the following function, find a point of maxima and a point of minima, if these exist

f(x) = 12x6 – 4x4 + 15x3 –1.

if you use math and /math in [] and this code you will get the function below.
f(x)=12x^6-4x^4+15x^3-1

I asuume you mean

$f(x)=12x^6-4x^4+15x^3-1$

$\frac{df}{dx}=72x^5-16x^3+45x^2=x^2(72x^3-16x+45)$

now lets focus on

$g(x)=72x^3-16x+45$

well

$g(-1)=-72+16+45=-11 \mbox{ and }g(0)=45$

so we know that it has at least one real root inbetwwen -1 and 0

if it is rational you can use the rational roots theorem, if not try numerical methods.

I hope this will get you started. Good luck.
• Apr 8th 2008, 12:31 PM
rajdey1
Quote:

Originally Posted by Moo
hello,

ya but please help me to move forward with the problem after frist derivative test i have been unable to locate the critical points
• Apr 8th 2008, 12:51 PM
Mathstud28
I will give you a hint
Quote:

Originally Posted by rajdey1
For the following function, find a point of maxima and a point of minima, if these exist

f(x) = 12x6 – 4x4 + 15x3 –1.

please help to find all the critical point of the function i am finding it difficult to solve the problem after doing frist derivative test

$f'(x)=72x^5-16x^3+15x^2$...here is your hint $f'(-1)=-11$ and $f'\bigg(\frac{-1}{2}\bigg)=-11$