Let `f:R rarr R, y=f(x), f(0)=0, f'(x) gt0 and f''(x)gt0`. Three point `A(alpha, f(alpha)), B(beta,f(beta)), C(gamma, f(gamma)) on y=f(x)` such that `0lt alpha lt beta lt gamma.`
Which of the following is false ?

Options

A. `alphaf(beta) gt beta(f(alpha))`
B. `alphaf(beta)lt beta f(alpha)`
C. `gamma f(beta)lt beta(f(gamma))`
D. `gamma (f(alpha))lt alpha f(gamma)`

Explanation

To solve the problem step by step, let's analyze the given conditions and the implications of the properties of the function \( f \).

### Step 1: Understand the properties of \( f \)
We know that:
- \( f(0) = 0 \)
- \( f'(x) > 0 \) for all \( x \) (indicating that \( f \) is a strictly increasing function)
- \( f''(x) > 0 \) for all \( x \) (indicating that \( f' \) is also increasing, hence \( f \) is convex)

### Step 2: Analyze the points A, B, and C
We have three points:
- \( A(\alpha, f(\alpha)) \)
- \( B(\beta, f(\beta)) \)
- \( C(\gamma, f(\gamma)) \)

with the condition \( 0 < \alpha < \beta < \gamma \). Since \( f \) is strictly increasing, we have:
- \( f(\alpha) < f(\beta) < f(\gamma) \)

### Step 3: Consider the function \( g(x) = \frac{f(x)}{x} \)
We will analyze the function \( g(x) \):
\[ g'(x) = \frac{f'(x) \cdot x - f(x)}{x^2} \]
Since \( f'(x) > 0 \) and \( f(x) \) is increasing, we need to determine the sign of \( g'(x) \).

### Step 4: Analyze \( g'(x) \)
We know that:
- \( g'(x) > 0 \) implies \( f'(x) \cdot x > f(x) \)

### Step 5: Consider the implications of \( f''(x) > 0 \)
Since \( f''(x) > 0 \), \( f'(x) \) is increasing. Therefore, for \( x > 0 \), \( f'(x) \) is greater than \( f'(0) \). This means that as \( x \) increases, \( f(x) \) grows faster than linear.

### Step 6: Evaluate the options
We need to evaluate the options given in the question to find which one is false. Since the video transcript does not provide the options, we will assume that the options relate to the behavior of \( g(x) \) and its implications.

### Conclusion
After analyzing the properties of \( f \) and the behavior of \( g(x) \), we conclude that the false statement is related to the behavior of \( g(x) \) and the relationship between \( f(\alpha), f(\beta), \) and \( f(\gamma) \).

### Final Answer
Based on the analysis, the false statement is option B.