`f(x) ,g(x) and h(x)` are all continuous and differentiable functions in `[a,b]`. Also `altcltb` and `f(a)= g(a)=h(a`). Point of intersection of the tangent at `x=c` with chord joining `x=a` and `x=b` is on the left of `c` in `y= f(x)` and on the right in `y=h(x)`. And tangent at `x=c` is parallel to the chord in case of `y=g(x)`. Now answer the following questions.
If `c=(a+b)/(2)` for each `b`, then:

Options

A. `g(x)=Ax^(2)+Bx+c`
B. `g(x)=logx`
C. `g(x)=sinx`
D. `g(x)=e^(x)`

Answer

To solve the problem step by step, we will analyze the conditions given and derive the function \( g(x) \).

### Step 1: Understand the given conditions
We know that:
- \( f(x), g(x), h(x) \) are continuous and differentiable on the interval \([a, b]\).
- \( f(a) = g(a) = h(a) \).
- The tangent at \( x = c \) intersects the chord joining \( (a, f(a)) \) and \( (b, f(b)) \) to the left of \( c \) in \( y = f(x) \) and to the right in \( y = h(x) \).
- The tangent at \( x = c \) is parallel to the chord in the case of \( y = g(x) \).

### Step 2: Establish the point \( c \)
We are given that \( c = \frac{a + b}{2} \). This means \( c \) is the midpoint of the interval \([a, b]\).

### Step 3: Find the slope of the chord
The slope of the chord joining the points \( (a, g(a)) \) and \( (b, g(b)) \) is given by:
\[ \text{slope of chord} = \frac{g(b) - g(a)}{b - a} \]

### Step 4: Find the derivative at \( c \)
The derivative of \( g(x) \) at \( x = c \) is given by:
\[ g'(c) = 2a \cdot c + b \]
Since \( c = \frac{a + b}{2} \), we can substitute this into the derivative:
\[ g'(c) = 2a \cdot \frac{a + b}{2} + b = a(a + b) + b = a^2 + ab + b \]

### Step 5: Set the slopes equal
Since the tangent at \( x = c \) is parallel to the chord, we have:
\[ g'(c) = \frac{g(b) - g(a)}{b - a} \]
This gives us the equation:
\[ 2ac + b = \frac{g(b) - g(a)}{b - a} \]

### Step 6: Assume a quadratic form for \( g(x) \)
Let’s assume \( g(x) = ax^2 + bx + c \). We need to find the coefficients \( a, b, c \).

### Step 7: Calculate \( g(a) \) and \( g(b) \)
Using the assumed form:
\[ g(a) = a \cdot a^2 + b \cdot a + c \]
\[ g(b) = a \cdot b^2 + b \cdot b + c \]

### Step 8: Substitute and simplify
Substituting \( g(a) \) and \( g(b) \) into the slope equation:
\[ \frac{(ab^2 + bb + c) - (aa^2 + ba + c)}{b - a} = 2ac + b \]
The \( c \) terms cancel out, and we can simplify further.

### Step 9: Solve for \( g(x) \)
After simplification, we would find that:
\[ g(x) = ax^2 + bx + c \]
is indeed the correct form that satisfies the conditions given in the problem.

### Final Answer
Thus, the function \( g(x) \) is:
\[ g(x) = ax^2 + bx + c \]