For what value of λ is the function defined by f(x)= ll (x^ 2 -2 x ), & if x 0 4 x+1, & if x>0 . continuous at x = 0? What about continuity at x = 1?

It is given that f(x)= c (x^ 2 -2 x ), if x 0 4 x+1, if x>0 . Let f be continuous at x = 0, then, we get _ x 0^ - f(x) = _ x 0^ + f(x) = f(0) _ x 0^ - f(x) = _ x 0^ - ( ( x^2 - 2x ) ) = _ x 0^ - ( ( 0^2 - 2 0 ) ) = 0 _ x 0^ + f(x) = _ x 0^ + (4x + 1) = 4 0 + 1 = 1 _ x 0^ - f(x) _ x 0^ + f(x) Therefore, there is no value of for which f is continuous at x = 0 f(1) = 4x + 1 = 4 × 1 + 1 = 5 _ x 1 (4x + 1) = 4 × 1 + 1 = 5 Then _ x 1 f ( x ) = f (1) Therefore, for any values of , f is continuous at x = 1

For what value of λ is the function defined by f(x)= ll (x^ 2 -2 …

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Question

For what value of λ is the function defined by $f(x)=\left\{\begin{array}{ll} {\lambda\left(x^{2}-2 x\right),} & {\text { if } x \leq 0} \\ {4 x+1,} & {\text { if } x>0} \end{array}\right.$ continuous at x = 0? What about continuity at x = 1?

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Answer

It is given that $f(x)=\left\{\begin{array}{c} {\lambda\left(x^{2}-2 x\right), \text { if } x \leq 0} \\ {4 x+1, \text { if } x>0} \end{array}\right.$
Let f be continuous at x = 0, then, we get
$\mathop {\lim }\limits_{x \to {0^ - }} f(x) = \mathop {\lim }\limits_{x \to {0^ + }} f(x) = f(0)$
$\mathop {\lim }\limits_{x \to {0^ - }} f(x) = \mathop {\lim }\limits_{x \to {0^ - }} \left( {\lambda \left( {{x^2} - 2x} \right)} \right) = \mathop {\lim }\limits_{x \to {0^ - }} \left( {\lambda \left( {{0^2} - 2 \times 0} \right)} \right) = 0$
$\mathop {\lim }\limits_{x \to {0^ + }} f(x) = \mathop {\lim }\limits_{x \to {0^ + }} (4x + 1)$ = 4 $\times$ 0 + 1 = 1
$\therefore \mathop {\lim }\limits_{x \to {0^ - }} f(x) \ne \mathop {\lim }\limits_{x \to {0^ + }} f(x)$
Therefore, there is no value of $\lambda$ for which f is continuous at x = 0
f(1) = 4x + 1 = 4 × 1 + 1 = 5
$\mathop {\lim }\limits_{x \to 1} (4x + 1)$ = 4 × 1 + 1 = 5
Then $\mathop {\lim }\limits_{{\mathbf{x}} \to 1} {\text{f}}({\text{x}}) = {\text{f}}(1)$
Therefore, for any values of $\lambda$, f is continuous at x = 1

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