A cantilever beam is shown in figure. Neglecting mass of beam, which one of the following options is correct representation of shear force diagram and bending moment diagram?

The correct option is the first option, which represents:
- A Shear Force Diagram (SFD) with a constant value of $P$ from the fixed support to the midpoint ($x=0$ to $x=\frac{L}{2}$), and $0$ from the midpoint to the free end ($x=\frac{L}{2}$ to $x=L$).
- A Bending Moment Diagram (BMD) that starts at $0$ at the fixed support, increases linearly to $\frac{PL}{2}$ at the midpoint, and remains constant at $\frac{PL}{2}$ from the midpoint to the free end.

Step-by-Step Explanation:

1. Determination of Support Reactions:
Let the fixed support at the left end be point $A$ ($x=0$), the point load $P$ be at the midpoint $B$ ($x=\frac{L}{2}$), and the free end with the concentrated counter-clockwise moment $M=\frac{PL}{2}$ be point $C$ ($x=L$).

Let $R_A$ be the vertical reaction force acting upwards at the fixed support $A$, and $M_A$ be the reaction moment (assumed clockwise) at $A$.

From vertical equilibrium of the beam:
$R_A-P=0\Rightarrow R_A=P$

From moment equilibrium about point $A$ (taking counter-clockwise moments as positive):
$-M_A-P\left(\frac{L}{2}\right)+M=0$

Substitute the value of the applied moment at the free end, $M=\frac{PL}{2}$:
$-M_A-\frac{PL}{2}+\frac{PL}{2}=0\Rightarrow M_A=0$

Thus, the reaction moment at the fixed support $A$ is exactly zero.

2. Shear Force (SF) and Bending Moment (BM) Functions:

For Region 1: $0\le x<\frac{L}{2}$ (between support $A$ and point load $B$):
- The only vertical force acting to the left of the section is the reaction force $R_A=P$ upwards. Therefore, the shear force is:
$V\left(x\right)=R_A=P$
- The bending moment at any point $x$ is:
$M\left(x\right)=R_A·x-M_A=Px-0=Px$
At $x=0$, $M\left(0\right)=0$.
At $x=\frac{L}{2}$, $M\left(\frac{L}{2}\right)=\frac{PL}{2}$.

For Region 2: $\frac{L}{2}<x\le L$ (between point load $B$ and free end $C$):
- The vertical forces to the left of the section are $R_A=P$ upwards and the point load $P$ downwards. Therefore, the shear force is:
$V\left(x\right)=R_A-P=P-P=0$
- The bending moment at any point $x$ is:
$M\left(x\right)=R_A·x-P\left(x-\frac{L}{2}\right)=Px-Px+\frac{PL}{2}=\frac{PL}{2}$
This shows that the bending moment remains constant at $\frac{PL}{2}$ throughout this region.

Conclusion:
Comparing these equations to the options, we find that the diagram showing a constant shear force of $P$ in the left half and $0$ in the right half, along with a bending moment that increases linearly from $0$ to $\frac{PL}{2}$ and then remains constant, is the correct representation.