Largest integer coefficient for a quadratic with no real roots, its vertex, a normal line and an enclosed area

The function \(f\) is defined by \(f(x)=x^2+kx+14\), where \(x\in\mathbb{R}\) and \(k\) is a positive integer.

1. Given that the graph of \(y=f(x)\) does not meet the \(x\)-axis, show that the largest possible value of \(k\) is \(7\).[2]

From this point onwards, take \(k=7\).

2. 1. Write down the equation of the line of symmetry of the graph of \(y=f(x)\).
   2. Hence, or otherwise, find the coordinates of the vertex of the graph of \(y=f(x)\).[3]

The line \(N\) is the normal to the graph of \(y=f(x)\) at the point where \(x=-2\). The diagram below shows part of the graph of \(y=f(x)\) and the line \(N\). The shaded region is enclosed by the graph, the line \(N\) and the \(y\)-axis.

3. Show that \(N\) has equation \(y=-\frac13x+\frac{10}{3}\).[5]
4. Hence, find the area of the shaded region.[5]