I have a question and I've made an error somewhere I'm sure, I was hoping somebody could guide me through the steps? :3
The first part is to solve (1+2x)^4
For this I got 1 + 8x + 24x^2 + 48x^3 + 16x^4
Then it asks to solve (1+2x)^4 + (1-2x)^4
And it's a prove question so it shows the answer as 2 + 48x^2 + 32x^4 (which I got)
I'm stuck with this:
Hence show that the curve y = (1+2x)^4 + (1-2x)^4 has only one stationary point, and state then co-ordinates.
I did f'(x) = 96x + 128x^3
And set equal to 0
And simplified to 0 = 3 + 4x^2
But how do I prove this is the only solution because I can't simply it any more or get the co-ordinates? I'm not allowed to use imaginary numbers because this is for a core 2 exam, any help is much appreciated :)
The first part is to solve (1+2x)^4
For this I got 1 + 8x + 24x^2 + 48x^3 + 16x^4
Then it asks to solve (1+2x)^4 + (1-2x)^4
And it's a prove question so it shows the answer as 2 + 48x^2 + 32x^4 (which I got)
I'm stuck with this:
Hence show that the curve y = (1+2x)^4 + (1-2x)^4 has only one stationary point, and state then co-ordinates.
I did f'(x) = 96x + 128x^3
And set equal to 0
And simplified to 0 = 3 + 4x^2
But how do I prove this is the only solution because I can't simply it any more or get the co-ordinates? I'm not allowed to use imaginary numbers because this is for a core 2 exam, any help is much appreciated :)
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You are right, you made a mistake somewhere in the beginning as 3+4x^2 has no real roots. So lets binomial expand this sucker. (1+2x)^4 =
1 + 8x + 24x^2 + 32x^3 + 16x^4
Ur mistake was on the fourth term right up there. You multiplied wrong. Did you get pascals triangle thing right? it should be 1 4 6 4 1 as the one you want.
Anyways, from there lets continue. Lets expand out (1-2x)^4 :
1 - 8x + 24x^2 -32x^3 + 16x^4
Now we add them. (Waves algebra wand) I got: 2 + 48x^2 + 32x^4 Which is exactly what we needed. Sweet! Now its time for derivatives. Derive that sucka! We get:
96x+128x^3 = 0
So, that was pointless because i got the same stuff as you, and I'm stuck also :/.
Wait a second, if we put 96x + 128x^3 equal to zero you can factor out an 8x making 8x(12+16x^2). The inside (12+16x^2) has no real roots, so you should only focus on the 8x part. When does 8x=0? Obviously when x=0!!! So now we have the x co-ordinate. Plug it into the original equation, we get the y coordinate as 2. So your answer should be: Stationary point at (0,2).
I decided to test this out and see if im right in a graphing calculator, and it works! (0,2) is the stationary point! Took me a while to figure out, but we got it!
1 + 8x + 24x^2 + 32x^3 + 16x^4
Ur mistake was on the fourth term right up there. You multiplied wrong. Did you get pascals triangle thing right? it should be 1 4 6 4 1 as the one you want.
Anyways, from there lets continue. Lets expand out (1-2x)^4 :
1 - 8x + 24x^2 -32x^3 + 16x^4
Now we add them. (Waves algebra wand) I got: 2 + 48x^2 + 32x^4 Which is exactly what we needed. Sweet! Now its time for derivatives. Derive that sucka! We get:
96x+128x^3 = 0
So, that was pointless because i got the same stuff as you, and I'm stuck also :/.
Wait a second, if we put 96x + 128x^3 equal to zero you can factor out an 8x making 8x(12+16x^2). The inside (12+16x^2) has no real roots, so you should only focus on the 8x part. When does 8x=0? Obviously when x=0!!! So now we have the x co-ordinate. Plug it into the original equation, we get the y coordinate as 2. So your answer should be: Stationary point at (0,2).
I decided to test this out and see if im right in a graphing calculator, and it works! (0,2) is the stationary point! Took me a while to figure out, but we got it!