Question

Given $f\left(x\right)\{\begin{array}{cc}|x-2|+\lambda & ifx\le 2\\ x^2+1& \text{if}x>2\end{array}$
If $f\left(x\right)$ has a local minimum at $x=2$, then which of the following is most appropriate

• A. $\lambda \le 5$
• B. $\lambda \ge 5$
• C. $\lambda \le 0$
• D. $\lambda \ge 0$

Answer 1

The correct option is A $\lambda \le 5$

$f\left(x\right)=\{\begin{array}{c}|x-2|+\lambda ;ifx\le 2\\ x^2+1;ifx>2\end{array}$

(Image)

$f\left(2\right)=|x-2|+\lambda =0+\lambda =\lambda$

$f(2-h)=|2-h-2|+\lambda =h+\lambda$

$f(2+h)={(2+h)}^2+1\Rightarrow 4+h^2+2h+1\Rightarrow 5+h^2+2h$

From graph

$f\left(2\right)\le f(2+h)$

$\lambda \le 5+h^2+2h$

$\lambda \le 5$ ($h=$relatively small)

$\lambda \le 5$ Answer