2. a) Determine the coordinates of all point(s) on the graph of y = 2x lnx where the tangent line is perpendicular to the line x+4y-7=0. b) Find the equation of the tangent line to the graph of f(x) = xe* + e* + 1 at the point where x = 1.

Part_a_ Determine the coordinates of all point(s) on the graph of $y=2x\ln x$ where the tangent line is perpendicular to the line $x+4y-7=0$.

Solution:-

1. Slope of line $x+4y-7=0$

Slope of line perpendicular to the line $ax+by+c=0$ is given as $m=\frac{b}{a}$

Substitute a=1 and b=4 in $m=\frac{b}{a}$

$m=\frac{4}{1}$

$m=4$ Equation 1

2. Slope of the tangent line of $y=2x\ln x$ is given as $\frac{dy}{dx}$

$\frac{dy}{dx}=\frac{d}{dx}\left(2x\ln x\right)$

$\frac{dy}{dx}=2\ln x\frac{d}{dx}\left(x\right)+2x\frac{d}{dx}\left(\ln x\right)$

$\frac{dy}{dx}=2\ln x+2x\times \frac{1}{x}$

$\frac{dy}{dx}=2\ln x+2$

Since, the tangent line is perpendicular to the line $x+4y-7=0$

Therefore, $\frac{dy}{dx}=4$

$4=2\ln x+2$

$2\ln x=4-2$

$2\ln x=2$

$\ln x=\frac{2}{2}$

$\ln x=1$

Take antilog in both sides

$x=e$ Equation 2

Substitute $x=e$ in $y=2x\ln x$

$y=2\times e\times \ln e$

$y=2e\times 1$

$y=2e$ Equation 2

Coordinates of the point $\left(e, 2e\right)$

Hence, the coordinates of all point on the graph of $y=2x\ln x$ where the tangent line is perpendicular to the line $x+4y-7=0$ is $\left(e, 2e\right)$