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

Neethu Nair, M.S. | Certified Educator

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We have to show that the curve y=2e^x+3x+5x^3  has no tangent line with slope 2.

Now, we know that the slope of the tangent is found by taking the first derivative of the curve. In other words,

\frac{dy}{dx}=2e^x+3+15x^2

Now, let us consider that there is a tangent line whose slope is 2.

To demonstrate,

\frac{dy}{dx}=2

2e^x+3+15x^2=2

For example, 2e^x+15x^2=-1

But we know that e^x  and x^2 are always positive. So the left hand side of the above equation will be always positive, but the right hand side is always negative.

Hence, there are no values of x for which we get the slope 2.

Therefore we can say that there are no tangent lines to the given curve with a slope of 2.

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## Related Questions

Rico Grant | Certified Educator

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We are asked to show that curve represented by the function y=2e^x+3x+5x^3  has no tangent line with a slope of 2:

The slope of a tangent line to a curve at a point is found by evaluating the first derivative of the function at that point.

Here the first derivative is:

y'=2e^x+3+15x^2

If we temporarily assume that...

(The entire section contains 4 answers and 604 words.)

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Inna Shpiro | Certified Educator

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