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Let f:R rarr R, y=f(x), f(0)=0, f'(x) gt...

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 true?

A

(a) `gammaf(gamma+beta-alpha)gt (gamma+beta-alpha)f(gamma)`

B

(b) `gammaf(gamma+beta-alpha)lt (gamma+beta-alpha)f(gamma)`

C

(c) `alphaf(gamma+beta-alpha)gt (gamma+beta-alpha)f(alpha)`

D

(d) None of these

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we start by analyzing the properties of the function \( f \) given in the question. 1. **Understanding the Function Properties**: - We know that \( f(0) = 0 \). - The first derivative \( f'(x) > 0 \) indicates that \( f(x) \) is an increasing function. This means that for any \( x_1 < x_2 \), \( f(x_1) < f(x_2) \). - The second derivative \( f''(x) > 0 \) indicates that \( f(x) \) is concave upward. This means that the slope of the function is increasing. 2. **Identifying Points A, B, and C**: - We have three points on the curve: \( A(\alpha, f(\alpha)) \), \( B(\beta, f(\beta)) \), and \( C(\gamma, f(\gamma)) \) with \( 0 < \alpha < \beta < \gamma \). 3. **Comparing Function Values**: - Since \( f \) is increasing, we can conclude that \( f(\alpha) < f(\beta) < f(\gamma) \). This is because \( \alpha < \beta < \gamma \) implies \( f(\alpha) < f(\beta) \) and \( f(\beta) < f(\gamma) \). 4. **Using the Concavity**: - The fact that \( f''(x) > 0 \) (the function is concave upward) implies that the rate of increase of \( f \) is itself increasing. This means that the difference \( f(\beta) - f(\alpha) \) is less than \( f(\gamma) - f(\beta) \). In other words, the function grows faster as \( x \) increases. 5. **Conclusion**: - From the properties of the function, we can conclude that: \[ f(\alpha) < f(\beta) < f(\gamma) \] - Therefore, the correct statement among the options provided must reflect this ordering. ### Final Answer: The correct option is that \( f(\alpha) < f(\beta) < f(\gamma) \).
To solve the problem, we start by analyzing the properties of the function \( f \) given in the question. 1. **Understanding the Function Properties**: - We know that \( f(0) = 0 \). - The first derivative \( f'(x) > 0 \) indicates that \( f(x) \) is an increasing function. This means that for any \( x_1 < x_2 \), \( f(x_1) < f(x_2) \). - The second derivative \( f''(x) > 0 \) indicates that \( f(x) \) is concave upward. This means that the slope of the function is increasing. 2. **Identifying Points A, B, and C**: ...
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