Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as : $f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4\end{cases}$ where $\mathrm{b} \in \mathbb{R}$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

Question

Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as :

$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4\end{cases}$

where $\mathrm{b} \in \mathbb{R}$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

$f$ is not differentiable at $x=4$

$f^{\prime}(3)+f^{\prime}(5)=\frac{35}{4}$

$f$ is increasing in $\left(-\infty, \frac{1}{8}\right) \cup(8, \infty)$

$f$ has a local minima at $x=\frac{1}{8}$

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