Question

Show that the curve \(y = 2{e^x} + 3x + 5{x^3}\) has no tangent line with slope 2.

Short answer

It is proved that the curve \(y = 2{e^x} + 3x + 5{x^3}\) has no tangent line with slope 2.

Step-by-step solution

 Tangent line to the curve

The tangent line to the curve \(y = f\left( x \right)\) at \(\left( {a,f\left( a \right)} \right)\) is the line through \(\left( {a,f\left( a \right)} \right)\) whose slope is equal to \(f'\left( a \right)\), the derivative of \(f\) at \(a\).

Show that the curve \(y = 2{e^x} + 3x + 5{x^3}\) has no tangent line

Differentiate the function as shown below:

\(\begin{align}y' &= \frac{d}{{dx}}\left( {2{e^x} + 3x + 5{x^3}} \right)\\ &= 2\frac{d}{{dx}}\left( {{e^x}} \right) + 3\frac{d}{{dx}}\left( x \right) + 5\frac{d}{{dx}}\left( {{x^3}} \right)\\ &= 2{e^x} + 3\left( 1 \right) + 5\left( 3 \right){x^2}\\ &= 2{e^x} + 3 + 15{x^2}\end{align}\)

Then \(2{e^x} > 0\) and \(15{x^2} \ge 0\), we obtain as shown below:

\(\begin{array}{c}y' > 0 + 3 + 0\\ = 3\end{array}\)

Hence, there is no tangent line with slope 2.

Thus, it is proved that the curve \(y = 2{e^x} + 3x + 5{x^3}\) has no tangent line with slope 2.