Chapter 3: Problem 24
If \(\mathrm{y}=|1-| 2-|3-| 4-\mathrm{x}|||| ;\) then number of points where \(\mathrm{y}\) is not differentiable; is (A) 1 (B) 3 (C) 5 (D) \(>5\)
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I.et \(\mathrm{f}(\mathrm{x})\) be a function such that \(\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y})\) and \(f(x)=\sin x g(x)\) for all \(x, y \in R\), If \(g(x)\) is a continuous function such that \(\mathrm{g}(0)=\mathrm{K}\), then \(f^{\prime}(\mathrm{x})\) is equal to (A) \(\mathrm{K}\) (B) \(\mathrm{Kx}\) (C) \(\mathrm{Kg}(\mathrm{x})\) (D) none
Let $f(x)=\left\\{\begin{array}{cl}\frac{\sin \left|x^{2}-5 x+6\right|}{x^{2}-5 x+6}, & x \neq 2,3 \\ 1 & , x=2 \text { or } 3\end{array}\right.$ The set of all points where \(f\) is differentiable is (A) \((\infty, \infty)\) (B) \((-\infty, \infty)-\\{2\\}\) (C) \((-\infty, \infty)-\\{3\\}\) (T) \((-\infty, \infty)-\\{2,3\\}\)
The number of points where \(f(x)=(x+1)^{23}+|x-1|^{\sqrt{3}}\), is non- differentiable is (A) 1 (B) 2 (C) 3 (D) none
Given \(\mathrm{f}(\mathrm{x})\) is a differentiable function of \(\mathrm{x}\), satisfying \(f(x) . f(y)=f(x)+f(y)+f(x y)-2\) and that \(f(2)=5\). Then \(\mathrm{f}(3)\) is equal to (A) 10 (B) 24 (C) 15 (D) none
The function $f(x)=\frac{|x|-x\left(3^{1 / x}+1\right)}{3^{1 / x}-1}, x \neq 0, f(0)=0$ is (A) discontinuous at \(x=0\) (B) continuous at \(\mathrm{x}=0\) but not differentiable there (C) both continuous and differentiable at \(\mathrm{x}=0\) (D) differentiable but not continuous at \(\mathrm{x}=0\)
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