On which two intervals is the function increasing?

Question

For x E [-11,11] the function f is defined by 

f(x)=x⁶(x-2)⁵

1- on which two intervals is the function increasing?

2- Find the region in which the function is positive:

3- Where does the function achieve its minimum?

 

Answer 1

First to find the coefficients quickly use pascal's triangle

1.             X

1.  1.     X+1

1 2. 1. X^2+2x+1

And so on.

So it's 1 3 3 1 for (x+1)^3

And 1 4 6 4 1 for (x+1)^4

And 1 5 10 10 5 1 for (x+1)^5

 

Since this is x-2 we distribute minus signs..

1 -5 10 -10 5 -1

And multiply through by increasing powers of two with 2^5

1 -5*2 10*4 -10*8 5*32 -32

Addthe exponents back in now:

X^5-10x^4+40x^3-80x^2+160x-32

 

and finally multiply through x^6

X^11-10x^10+40x^9-80x^8+160x^7-32x^6

 

Take the derivative here:

11x^10-100x^9+360x^8-640x^7+1020x^6.....

 

Now we know that 0 and 2 are roots so test one and negative one heRe to see where f is increasing.

We know that f is positive for x greater than two in this case looking at the function.

 

As for the minimum, test to see where f begins to increase (0 or two) and that will be your minimum,