Find the point $P$ situated on the plot of the function $f(x) = e^x + x$, where the tangent to the plot passes through the origin.
Here is my approach:
Let $d: y - f(x_0) = f'(x_0)(x - x_0)$ be the equation of the tangent to the function plot at point $M(x_0, f(x_0))$.
The tangent passes through the origin, which means that $y = xf'(0)$.
$f'(x) = e^x + 1$, we get $f'(0) = 2$.
So, the equation of the tangent to the function plot which passes through the origin is $d: y - 2x = 0$.
Now, we need to find the intersection of the function plot with this tangent line. From the equation of the tangent we get $y = 2x$, so $f(x) = 2x$.
Plugging what we've just got into $f(x) = e^x + x$, we get $e^x - x = 0$, which has no real solutions.
So, the point P doesn't exist.
Is the solution I provided a correct one? Are there any mistakes?
Thank you in advance!