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Let of be a twice differentiable function on R. Given that f^ prime prime (x) > 0 for all x \in R then
(a) f(x) = 0 has exactly two solutions in R
(d) f(x) = 0 has positive solution if f(0) = 0 and f' * (0) = 0
(c) f(x) = 0 has no positive solution f(0) = 0 and f' * (0) > 0 (d) f(x) = 0 has no positive solution if f(0) = 0 and f' * (0) < 0?="" />
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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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Let of be a twice differentiable function on R. Given that f^ prime prime (x) > 0 for all x \in R then(a) f(x) = 0 has exactly two solutions in R(d) f(x) = 0 has positive solution if f(0) = 0 and f' * (0) = 0(c) f(x) = 0 has no positive solution f(0) = 0 and f' * (0) > 0 (d) f(x) = 0 has no positive solution if f(0) = 0 and f' * (0)
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Let of be a twice differentiable function on R. Given that f^ prime prime (x) > 0 for all x \in R then(a) f(x) = 0 has exactly two solutions in R(d) f(x) = 0 has positive solution if f(0) = 0 and f' * (0) = 0(c) f(x) = 0 has no positive solution f(0) = 0 and f' * (0) > 0 (d) f(x) = 0 has no positive solution if f(0) = 0 and f' * (0) for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let of be a twice differentiable function on R. Given that f^ prime prime (x) > 0 for all x \in R then(a) f(x) = 0 has exactly two solutions in R(d) f(x) = 0 has positive solution if f(0) = 0 and f' * (0) = 0(c) f(x) = 0 has no positive solution f(0) = 0 and f' * (0) > 0 (d) f(x) = 0 has no positive solution if f(0) = 0 and f' * (0) covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let of be a twice differentiable function on R. Given that f^ prime prime (x) > 0 for all x \in R then(a) f(x) = 0 has exactly two solutions in R(d) f(x) = 0 has positive solution if f(0) = 0 and f' * (0) = 0(c) f(x) = 0 has no positive solution f(0) = 0 and f' * (0) > 0 (d) f(x) = 0 has no positive solution if f(0) = 0 and f' * (0) .
Solutions for Let of be a twice differentiable function on R. Given that f^ prime prime (x) > 0 for all x \in R then(a) f(x) = 0 has exactly two solutions in R(d) f(x) = 0 has positive solution if f(0) = 0 and f' * (0) = 0(c) f(x) = 0 has no positive solution f(0) = 0 and f' * (0) > 0 (d) f(x) = 0 has no positive solution if f(0) = 0 and f' * (0) in English & in Hindi are available as part of our courses for UPSC. Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of Let of be a twice differentiable function on R. Given that f^ prime prime (x) > 0 for all x \in R then(a) f(x) = 0 has exactly two solutions in R(d) f(x) = 0 has positive solution if f(0) = 0 and f' * (0) = 0(c) f(x) = 0 has no positive solution f(0) = 0 and f' * (0) > 0 (d) f(x) = 0 has no positive solution if f(0) = 0 and f' * (0) defined & explained in the simplest way possible. Besides giving the explanation of Let of be a twice differentiable function on R. Given that f^ prime prime (x) > 0 for all x \in R then(a) f(x) = 0 has exactly two solutions in R(d) f(x) = 0 has positive solution if f(0) = 0 and f' * (0) = 0(c) f(x) = 0 has no positive solution f(0) = 0 and f' * (0) > 0 (d) f(x) = 0 has no positive solution if f(0) = 0 and f' * (0) , a detailed solution for Let of be a twice differentiable function on R. Given that f^ prime prime (x) > 0 for all x \in R then(a) f(x) = 0 has exactly two solutions in R(d) f(x) = 0 has positive solution if f(0) = 0 and f' * (0) = 0(c) f(x) = 0 has no positive solution f(0) = 0 and f' * (0) > 0 (d) f(x) = 0 has no positive solution if f(0) = 0 and f' * (0) has been provided alongside types of Let of be a twice differentiable function on R. Given that f^ prime prime (x) > 0 for all x \in R then(a) f(x) = 0 has exactly two solutions in R(d) f(x) = 0 has positive solution if f(0) = 0 and f' * (0) = 0(c) f(x) = 0 has no positive solution f(0) = 0 and f' * (0) > 0 (d) f(x) = 0 has no positive solution if f(0) = 0 and f' * (0) theory, EduRev gives you an ample number of questions to practice Let of be a twice differentiable function on R. Given that f^ prime prime (x) > 0 for all x \in R then(a) f(x) = 0 has exactly two solutions in R(d) f(x) = 0 has positive solution if f(0) = 0 and f' * (0) = 0(c) f(x) = 0 has no positive solution f(0) = 0 and f' * (0) > 0 (d) f(x) = 0 has no positive solution if f(0) = 0 and f' * (0) tests, examples and also practice UPSC tests.
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