Let of be a twice differentiable function on R. Given that f^ prime pr...
Understanding the Implications of f''(x) > 0
Given that the second derivative of a function f, f''(x) > 0 for all x in R, indicates that the function is concave up everywhere. This has significant implications for the behavior of f and its first derivative f'.
Consequences of f''(x) > 0
- Increasing First Derivative: Since f''(x) > 0, the first derivative f'(x) is a strictly increasing function. This means that once f' becomes positive, it stays positive for all x > c, where c is any point where f' is non-negative.
- Behavior at Critical Points: If f(0) = 0 and f'(0) = 0, then f'(x) will become positive for x > 0, indicating that f(x) will start increasing after 0. Hence, there will be no additional zeros after this point, leading to exactly two roots: one at x = 0 and another at some negative value.
Evaluating the Given Statements
- (a) f(x) = 0 has exactly two solutions in R: This is incorrect. There could be one solution or none based on the behavior of f.
- (b) f(x) = 0 has a positive solution if f(0) = 0 and f'(0) = 0: This is correct. The function will begin increasing after x = 0, thus allowing another root.
- (c) f(x) = 0 has no positive solution if f(0) = 0 and f'(0) > 0: This is correct. Since f starts positive and continues increasing, it will not return to zero.
- (d) f(x) = 0 has no positive solution if f(0) = 0 and f'(0) < 0:="" this="" is="" incorrect.="" the="" function="" could="" cross="" the="" x-axis="" again="" depending="" on="" its="" behavior="" before="" />
Conclusion
In summary, f''(x) > 0 leads to specific behaviors in f and f', influencing the number and nature of solutions to f(x) = 0. The key lies in the sign of f' at critical points.
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