JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let a function \(f: R \rightarrow R\) be defined as :
\(f(x)=\left\{\begin{array}{ll} \int_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4
\end{array}\right.\)
where \(b \in R\). If \(f\) is continuous at \(x=4\), then which of the following statements is NOT true?
- A \(f\) is not differentiable at \(x=4\)
- B \(f^{\prime}(3)+f^{\prime}(5)=\frac{35}{4}\)
- C \(f\) is increasing in \(\left(-\infty, \frac{1}{8}\right) \cup(8, \infty)\)
- D \(f\) has a local minima at \(x=\frac{1}{8}\)
Answer & Solution
Correct Answer
(C) \(f\) is increasing in \(\left(-\infty, \frac{1}{8}\right) \cup(8, \infty)\)
Step-by-step Solution
Detailed explanation
Given \(f(x)\left\{\begin{array}{ll}\int_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4\end{array}\right.\) \(f(x)\) is continuous at \(x=4\) So \(\lim _{x \rightarrow 4^{-}} f(x)=\lim _{x \rightarrow 4^{+}} f(x)=f(4)\) So…
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