The number of points, where the function $f: \mathbb{R} \to \mathbb{R}$, $$ f(x) = |x - 1|\cos|x - 2|\sin|x - 1| + (x - 3)|x^2 - 5x + 4|, $$ is NOT differentiable, is
To determine the number of points where the function f(x) = |x - 1|\cos|x - 2|\sin|x - 1| + (x - 3)|x^2 - 5x + 4| is not differentiable, we need to analyze it carefully.
The function is a piecewise combination of absolute value functions, products, and trigonometric functions, which can cause nondifferentiability at critical points. Let's break it down.
Consider the points at which the components of f(x) are not differentiable:
The expression |x - 1| is not differentiable at x = 1.
The expression |x - 2| is not differentiable at x = 2.
The expression |x^2 - 5x + 4| can be rewritten as |(x - 1)(x - 4)|, which is not differentiable at x = 1 and x = 4.
Thus, the points of nondifferentiability are at x = 1, x = 2, and x = 4. In total, there are three points where f(x) is not differentiable.
Let's match this with the options provided:
Option 1: 1.14 — Incorrect. Does not match our result.
Option 2: 1.78 — Incorrect. Does not match our result.
Option 3: 2.34 — Incorrect. Does not match our result.
Option 4: 3.02 — Incorrect. Does not match our result.
However, given the correct choice should be related to the exact count, the expected correct numerical prefix would be "3" since we established it has three non-differentiable points. It appears there might be an error in transcribing the options or the correct answer provided does not match our logical deduction. Please verify the question options and avoid discrepancies.
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Molarity of resultant solution is given by Molarity=\(\frac{10}{600}\) of equivalents [H+]=\(\frac{10}{600}\) pH=−log[H+] pH=−log[\(\frac{10}{600}\)] pH=1.778 So, the correct option is (B): 1.78
The primary and secondary amines, including several amine derivatives, have a direct impact on their properties due to the presence of hydrogen bonding. The compounds containing phosphorus have a lower boiling point and the compounds containing amines and alcohol have a higher boiling point. The structure of alkanols is immensely similar to that of amine except the presence of the hydroxyl group. In such a case, oxygen has a higher electronegativity than that of nitrogen, so alkanol compounds are more acidic in nature in comparison to the amines.
On account of the ability to form hydrogen bonds, the amines have tendencies of high solubility in water. The amine molecules such as Ethyl, diethyl, triethyl, and Methyl are gaseous in nature. Whereas, higher weight amines have a solid structure and alkyl amines have a liquid structure. There is an ammonia smell to gaseous amines and a fishy smell to liquid amines. The solubility of amines entirely depends upon the number of carbon atoms in the molecule.
