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C3- stuck on a question

Curve C has equation

y=e^{2x}+x^2+4x+1

Show that the x-coordinate of any stationary point of C satisfies the equation.

x=-2-e^2x

I differentiated the equation, then equated it to 0 and got

2e^{2x}+2x+4=0

I assume I have to solve for x, but I get stuck at

e^{2x}+x=\frac{e-4}{2}

Could anyone help?

Reply 1

Notnek

Original post by joyoustele

Curve C has equation

y=e^{2x}+x^2+4x+1

Show that the x-coordinate of any stationary point of C satisfies the equation.

x=-2-e^2x

I differentiated the equation, then equated it to 0 and got

2e^{2x}+2x+4=0

I assume I have to solve for x, but I get stuck at

e^{2x}+x=\frac{e-4}{2}

Could anyone help?

I'm not sure how you ended up with that. You don't need to "solve for x" here, instead just rearrange your equation to get to their one. So you have

2e^{2x}+2x+4=0

and you need

x=-2-e^{2x}

Start by moving the

2e^{2x}

and the

4

to the other side.

Reply 2

joyoustele

Original post by Notnek

I'm not sure how you ended up with that. You don't need to "solve for x" here, instead just rearrange your equation to get to their one. So you have

2e^{2x}+2x+4=0

and you need

x=-2-e^{2x}

Start by moving the

2e^{2x}

and the

4

to the other side.

Oh, I just didnt understand the Question. So if a question was to say show that something x satisfies [equation], do I just always rearrange to make their equation?

Reply 3

etothepiiplusone

Original post by joyoustele

Oh, I just didnt understand the Question. So if a question was to say show that something x satisfies [equation], do I just always rearrange to make their equation?

Yes.

What else could you do?

Reply 4

amzyrads

Original post by joyoustele

Curve C has equation

y=e^{2x}+x^2+4x+1

Show that the x-coordinate of any stationary point of C satisfies the equation.

x=-2-e^2x

I differentiated the equation, then equated it to 0 and got

2e^{2x}+2x+4=0

I assume I have to solve for x, but I get stuck at

e^{2x}+x=\frac{e-4}{2}

Could anyone help?

dy/dx= 2e^2x+ 2x+ 2
at stationary point dy/dx=0 therefore
2e^2x+2x+4=0
e^2x+x+2=0
x= -2-e^2x
is that right?

Reply 5

Notnek

Original post by joyoustele

Oh, I just didnt understand the Question. So if a question was to say show that something x satisfies [equation], do I just always rearrange to make their equation?