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The slope of a line tangent to the curve y = f(x) at a point where x = a, is given by the value of f'(a).
In the problem, the curve is represented by y = 2*e^x + 5x^3 + 3x. First determine the derivative `(dy)/(dx)` .
`(dy)/(dx)= (2*e^x + 5x^3 + 3x)'`
= `2*e^x + 15x^2 + 3`
If a tangent to this curve has slope 2, the equation 2*e^x + 15x^2 + 3 = 2 should have a real root.
2*e^x + 15x^2 + 3 = 2
=> 2*e^x + 15x^2 = 2 - 3 = -1
As e is positive, for no real value of x is the value of e^x negative. Similarly, x^2 is positive for all real values of x. The value of 2*e^x + 15x^2 is positive for all real values of x.
Therefore, the equation 2*e^x + 15x^2 = -1 does not have a real root. As a result, the slope of a line tangent to the given curve cannot be equal to 2.
This proves that the curve y = 2*e^x +3x + 5x^3 has no tangent line with slope 2.
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