Statement-1: Let f(x) and g(x) be two real functions connected by the relation
`g(x)=f(x)-2 (f(x))^2+4(f(x))^3`
Then f(x) and g(x) increase or decrease together .
Statement-2: if `b^2 -4ac lt 0 and a gt 0 then ax^2 + bx +c gt 0 "for all statement -2 " x in R`

Options

A. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is a correct explanation for Statement-8
B. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is not a correct explanation for Statement-8
C. Statement-1 True statement -1 is True,Statement -2 is False
D. Statement-1 is False ,Statement -2 is True

Explanation

To solve the given problem, we need to analyze both statements and determine their validity step by step.

### Step 1: Analyzing Statement 1
We have the relation:
\[ g(x) = f(x) - 2(f(x))^2 + 4(f(x))^3 \]

To understand how \( f(x) \) and \( g(x) \) behave, we can rewrite \( g(x) \):
\[ g(x) = f(x) \left( 1 - 2f(x) + 4(f(x))^2 \right) \]

Let \( y = f(x) \). Then we can express \( g(x) \) in terms of \( y \):
\[ g(y) = y \left( 1 - 2y + 4y^2 \right) \]

### Step 2: Finding the Critical Points
Next, we need to analyze the quadratic expression:
\[ h(y) = 4y^2 - 2y + 1 \]

To find the critical points, we can calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-2)^2 - 4 \cdot 4 \cdot 1 = 4 - 16 = -12 \]

Since \( D < 0 \), the quadratic \( h(y) \) has no real roots and opens upwards (since the coefficient of \( y^2 \) is positive). Therefore, \( h(y) > 0 \) for all \( y \in \mathbb{R} \).

### Step 3: Conclusion for Statement 1
Since \( h(y) > 0 \) for all \( y \), it implies that \( g(y) \) retains the same sign as \( f(y) \). Thus, if \( f(x) \) is increasing, \( g(x) \) is also increasing, and if \( f(x) \) is decreasing, \( g(x) \) is also decreasing. Therefore, Statement 1 is true.

### Step 4: Analyzing Statement 2
Statement 2 states:
If \( b^2 - 4ac < 0 \) and \( a > 0 \), then \( ax^2 + bx + c > 0 \) for all \( x \in \mathbb{R} \).

This is a standard result in quadratic functions. The condition \( b^2 - 4ac < 0 \) indicates that the quadratic does not intersect the x-axis (i.e., it has no real roots), and since \( a > 0 \), the parabola opens upwards. Therefore, the quadratic is always positive for all \( x \in \mathbb{R} \).

### Step 5: Conclusion for Statement 2
Since both conditions of Statement 2 are satisfied, Statement 2 is also true.

### Final Conclusion
Both statements are true. Therefore, the answer is that both statements are correct.