Find the values of a and b, if the function f defined by f(x) = x… - Vidyadip

Question

Find the values of a and b, if the function f defined by
$\text{f}(x) = \begin{cases} x^{2} + 3x + \text{a} \text{ }, & x \leq 1\\ bx + 2 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ },& x > 1\\ \end{cases}$
is differentiable at x = 1.

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Answer

$\text{f}'_{1-} = \text{2x + 3 = 5}$
$\text{f}'_{1+} = \text{b}$
$\text{f}_{1-} = \text{f}'_{1+} \Rightarrow \text{b = 5}$
$\lim\limits_{\text{x} \rightarrow 1^{-}} \text{f(x)} = \text{f(1)} = \lim\limits_{\text{x} \rightarrow 1^{+}} \text{f(x)}$
$\Rightarrow \text{4 + a = b + 2}$
$\Rightarrow \text{a = 3}$

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