Question:
Let \(f:[0,2] \rightarrow R\) be the function defined by
\(f(x)=(3-\sin (2 \pi x)) \sin \left(\pi x-\frac{\pi}{4}\right)-\sin \left(3 \pi x+\frac{\pi}{4}\right)\) If \(\alpha, \beta \in[0,2]\) are such that \(\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]\), then the value of \(\beta-\alpha\) is ______
Let \(f:[0,2] \rightarrow R\) be the function defined by
\(f(x)=(3-\sin (2 \pi x)) \sin \left(\pi x-\frac{\pi}{4}\right)-\sin \left(3 \pi x+\frac{\pi}{4}\right)\) If \(\alpha, \beta \in[0,2]\) are such that \(\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]\), then the value of \(\beta-\alpha\) is ______
\(f(x)=(3-\sin (2 \pi x)) \sin \left(\pi x-\frac{\pi}{4}\right)-\sin \left(3 \pi x+\frac{\pi}{4}\right)\) If \(\alpha, \beta \in[0,2]\) are such that \(\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]\), then the value of \(\beta-\alpha\) is ______
Updated On: May 22, 2024
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Correct Answer: 1
Solution and Explanation
\(\text{The value of}\) \(\beta-\alpha\) is \(\underline{1}.\)
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Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.