Question

Let $f\left(x\right)=\{\begin{array}{c}(x-1{)}^2+4x<1\\ a+x^{2}x\ge 1\end{array}$

• A. If $\left(x\right)$ is continuous $\forall x\epsilon R$, then $a=3$
• B. If $\left(x\right)$ is differentiable $\forall x\epsilon R$, then $a=3$
• C. If $\left(x\right)$ has a local minima at $x=1$, then $a\le 3$
• D. If $a=4$ & $\left(x\right)=k$ has exactly one solution, then k $\epsilon (4,5]$

Answer 1

Answer: (A), (C)

If $\left(x\right)$ is continuous $\forall x\epsilon R$, then $a=3$
If $\left(x\right)$ has a local minima at $x=1$, then $a\le 3$

(A) For continuity $f(1^{)}=f(1^{+})\equiv f(1)$
$\Rightarrow 4=a+1\Rightarrow a=3$
(B) $\because$ f(x) is differentiable
$\Rightarrow f^{\prime }\left(1\right)=f^{\prime }\left(1^{+}\right)\Rightarrow 2\left(x\right){|}_{x=1}=2.1$ which is not possible.