Show that the curve y = 2e^x + 3x + 5x^3 has no tangent line with slope 2.

So far, I've set it's derivative to 2... now I'm tempted to take natural logs, but it doesn't seem to lead me anywhere

Question

Show that the curve y = 2e^x + 3x + 5x^3 has no tangent line with slope 2.  

So far, I've set it's derivative to 2... now I'm tempted to take natural logs, but it doesn't seem to lead me anywhere

zepdrix · Answer

$$\Large y\quad=\quad  2e^x+3x+5x^3$$$$\Large y'\quad=\quad 2e^x+3+15x^2\quad=\quad 2$$So this is how far you've gotten, yes?
And now we need to show that there are no values of x that will satisfy this.
Ok ummm...

anonymous · Answer

Correct

anonymous · Answer

At least, that's my interpretation too

zepdrix · Answer

If we rearrange things...$$\Large -2e^x\quad=\quad 1+15x^2$$
We can think of it like this I suppose, as two separate functions, one on each side.
On the right side we have a parabola opening upward from y=1.
On the left side we have an exponential function which never goes above y=0.

So there is no point of intersection. No solution for x.

zepdrix · Answer

I don't think we can much better than that :(
You can't solve for x explicitly since we have the exponential and polynomial x.

loser66 · Answer

I wonder why we have to do more? 3 > 2 , x^2> 0 with any of x, e^x never <0 , the sum of them cannot be 2. It's trivial

anonymous · Answer

Thanks to both of you

goformit100 · Answer

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