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Let , and exists. The statement “ is continuous at ” is true. Justify. Hence, find a relation…

Mathematics20256 marksLong answer
Let $f(x) = \begin{cases} a + \sin^{-1}(x+b), & x \ge 1 \\ x, & x < 1 \end{cases}$, and $f'(1)$ exists.
(a)
The statement “$f(x)$ is continuous at $x = 1$” is true. Justify.
(b)
Hence, find a relation between $a$ and $b$.
(c)
Find $f'(x)$.
(d)
Hence, find the values of $a$ and $b$.

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Continuity, Differentiability and Differentiation

From ISC 2025 Practice Mathematics, question 131.