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Let f(x) = cos(|pi - x|) + (x - pi) * sin |x| and g(x) = x ^ 2 for x \in R If h(x) = f(g(x)) then
(a) h is not differentiable at x = 0
(c) h^ prime prime (x) = 0 has a solution in (- pi, pi)
(d) There exists x_{0} \in (- pi, pi) such that h(x_{0}) = x_{0}
(b) h' * (sqrt(pi)) = 0?
Most Upvoted Answer
Let f(x) = cos(|pi - x|) + (x - pi) * sin |x| and g(x) = x ^ 2 for x \...
Function Definitions
- f(x) = cos(|pi - x|) + (x - pi) * sin |x|
- g(x) = x^2 for x ∈ R
- h(x) = f(g(x)) = f(x^2)
Analysis of h(x)
- To understand h(x), analyze f(x) at g(x):
- g(x) = x^2 is always non-negative.
- Thus, h(x) = f(x^2) = cos(|pi - x^2|) + (x^2 - pi) * sin |x^2|.
Points to Consider
- (a) h is not differentiable at x = 0:
- At x = 0, g(0) = 0 and f(0) needs careful evaluation. The piecewise nature of f(x) may lead to a cusp, indicating non-differentiability.
- (b) h' * (sqrt(pi)) = 0?:
- To determine this, find h'(x) = f'(g(x)) * g'(x). Since g'(x) = 2x, evaluating at x = sqrt(pi) may yield zero if f'(g(sqrt(pi))) = f'(pi) is zero.
- (c) h''(x) = 0 has a solution in (-pi, pi):
- This requires the second derivative to be analyzed. Considering the oscillatory nature of f(x), it’s plausible h''(x) = 0 has solutions in the interval.
- (d) There exists x_0 in (-pi, pi) such that h(x_0) = x_0:
- By the Intermediate Value Theorem, since h(x) is continuous and varies from negative to positive values in the interval, there must exist a point where h(x_0) = x_0.
Conclusion
- Each statement is backed by the nature of f(x) and g(x), their interaction, and basic calculus principles. Further analysis of derivatives may provide deeper insights into the behavior of h(x) in the specified intervals.
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Let f(x) = cos(|pi - x|) + (x - pi) * sin |x| and g(x) = x ^ 2 for x \in R If h(x) = f(g(x)) then (a) h is not differentiable at x = 0(c) h^ prime prime (x) = 0 has a solution in (- pi, pi)(d) There exists x_{0} \in (- pi, pi) such that h(x_{0}) = x_{0}(b) h' * (sqrt(pi)) = 0?
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Let f(x) = cos(|pi - x|) + (x - pi) * sin |x| and g(x) = x ^ 2 for x \in R If h(x) = f(g(x)) then (a) h is not differentiable at x = 0(c) h^ prime prime (x) = 0 has a solution in (- pi, pi)(d) There exists x_{0} \in (- pi, pi) such that h(x_{0}) = x_{0}(b) h' * (sqrt(pi)) = 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 f(x) = cos(|pi - x|) + (x - pi) * sin |x| and g(x) = x ^ 2 for x \in R If h(x) = f(g(x)) then (a) h is not differentiable at x = 0(c) h^ prime prime (x) = 0 has a solution in (- pi, pi)(d) There exists x_{0} \in (- pi, pi) such that h(x_{0}) = x_{0}(b) h' * (sqrt(pi)) = 0? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x) = cos(|pi - x|) + (x - pi) * sin |x| and g(x) = x ^ 2 for x \in R If h(x) = f(g(x)) then (a) h is not differentiable at x = 0(c) h^ prime prime (x) = 0 has a solution in (- pi, pi)(d) There exists x_{0} \in (- pi, pi) such that h(x_{0}) = x_{0}(b) h' * (sqrt(pi)) = 0?.
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