Answer
See the explanation
Work Step by Step
Given: $y=2e^x+3x+5x^3$
Find $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{d}{dx}(2e^x+3x+5x^3)$ (Use the sum rule)
$\frac{dy}{dx}=\frac{d}{dx}(2e^x)+\frac{d}{dx}(3x)+\frac{d}{dx}(5x^3)$
$\frac{dy}{dx}=2e^x+3+15x^2$
Since for any real number $x$, $e^x>0$ and $x^2\geq 0$, we have
$2e^x+3+15x^2>2\cdot 0+3+15\cdot 0$
$2e^x+3+15x^2>3$
$\frac{dy}{dx}>3$
It means that the slope of any tangent line to the curve is greater than 3.
Consequently, the curve has no tangent line with the slope of 2.
