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(C) For $f(x)$ to be derivable at $x = c$,it must first be continuous at $x = c$.$1$. Continuity at $x = c$:$\lim_{x 	o c^-} f(x) = \lim_{x 	o c^+}

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$a = \cos c, b = \sin c - c \cos c$

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(C) For $f(x)$ to be derivable at $x = c$,it must first be continuous at $x = c$.$1$. Continuity at $x = c$:$\lim_{x 	o c^-} f(x) = \lim_{x 	o c^+} f(x) = f(c)$$\sin c = ac + b$  --- (Equation $1$)$2$. Differentiability at $x = c$:$f'(x) = \begin{cases} \cos x & 	ext{if } x  c \end{cases}$For $f'(c)$ to exist,the left-hand derivative must equal the right-hand derivative:$\lim_{x 	o c^-} f'(x) = \lim_{x 	o c^+} f'(x)$$\cos c = a$$3$. Substituting $a = \cos c$ into Equation $1$:$\sin c = (\cos c)c + b$$b = \sin c - c \cos c$Thus,$a = \cos c$ and $b = \sin c - c \cos c$.

$A$ function $f$ is defined as follows:
$f(x) = \begin{cases} \sin x & \text{if } x \le c \\ ax + b & \text{if } x > c \end{cases}$
where $c$ is a known quantity. If $f$ is derivable at $x = c$,then the values of $a$ and $b$ are . . . . . . and . . . . . . respectively.

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$A$ function $f$ is defined as follows:$f(x) = \begin{cases} \sin x & \text{if } x \le c \\ ax + b & \text{if } x > c \end{cases}$where $c$ is a known quantity. If $f$ is derivable at $x = c$,then the values of $a$ and $b$ are . . . . . . and . . . . . . respectively.