If the function $f$ defined by $ f(x)=\left\{\begin{array}{llc} \cos x, & \text { if } & x \leq 0 \\ 3…

If the function $f$ defined by $ f(x)=\left\{\begin{array}{llc} \cos x, & \text { if } & x \leq 0 \\ 3 x+\alpha, & \text { if } & 0 < x < 2 \\ \beta x+3, & \text { if } & 2 \leq x \leq 4 \\ 11, & \text { if } & x>4 \end{array}\right. $ where, $\alpha$ and $\beta$ are real constants, is continuous on $R$, then $\alpha^2+\beta^2=$
  1. 3
  2. 9
  3. 5
  4. 1

Solution

$f(x)$ is continuous on $R$, so $ \begin{array}{rlrl} \text { at } x=0, & \lim _{x \rightarrow 0^{+}} f(x) & =f(0) \\ \Rightarrow & 0+\alpha=\cos 0 & =1 \\ \alpha & =1 \\ \text { at } x=2, & \lim _{x \rightarrow 2^{-}} f(x) & =f(2) \\ 6+\alpha & =2 \beta+3 \end{array} $ $\begin{aligned} 7 & =2 \beta+3 \\ 2 \beta & =4 \Rightarrow \beta=2 \\ \text { So, } \quad \alpha^2+\beta^2 & =1+4=5 .\end{aligned}$

Asked in: AP EAMCET 2018 (24 Apr Shift 1)

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