00:01
Question with us and the question says that there is a function of x at the f of x and it has a constant a x raised to the power of four plus eight x squared.
00:11
The question further states that there are three turning points and it also has a maximum value of f of x at eight.
00:23
So at eight there is one of the turning points.
00:26
So question further states that there is an intercept, there is a zero.
00:30
There is a zero a root at 2 and hence since it's a root we will have f of x as 0.
00:38
So we'll key in that to find the value of a so we have 0 is equals to 16 a plus 32 we have a as minus 32 darby 16 that is minus 2 now the question wants us is to find further zeros of it now it can be roots and it can be the turning point set as well.
00:59
So for that we will move on with first the root so for the roots we have minus 2 x to the power of 4 plus 8 x squared is equal to 0 this would give us the roots so we'll take minus 2 x squared commons we left we are left with x square minus 4 is equal to 0 so we further factorize it we get minus 2 x square x plus 2 and x minus 2 is equal to 0 so to doing that, we get our zeros as minus 2x squared is equals to 0.
01:37
This becomes x is equals to 0.
01:40
We have got x plus 2 is equal to 0, which becomes x equals to minus 2.
01:44
And we have got x minus 2 is equals to 0, which becomes x is equals to 2.
01:49
Those are some zeros.
01:50
Now moving on towards the turning points, we'll have to differentiate f of x and minus 2x raised to the power of 4 plus 8x square.
02:01
Becomes minus 8x cubed plus 16x and since it's at a turning point the gradient would be equal to 0...