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

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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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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...

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