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

Answer 1

y= 2e^x + 3x +5x^3

1) take the derivative of your function with respect to x . .

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

2) derivative means slope of a tangent line so we can see that derivative ( slop of tangent line )

more than 2 .

y ' > 2

how do we know it is larger than to ?

2e^x >0 3>2 and 15x^2 > or = 0 so no mater what x value we plug in our y' value will be more than 2 .

Answer 2

y = 2e^x + 3x + 5x³

Slope of tangent = y' = 2e^x + 15x² + 3 > 3 since 2e^x > 0 and 15x² ≥ 0