हिंदी

If F (X) = | X − a | ϕ (X), Where ϕ (X) is Continuous Function, Then (A) F' (A+) = ϕ (A) (B) F' (A−) = −ϕ (A) (C) F' (A+) = F' (A−) (D) None of These - Mathematics

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प्रश्न

If f (x) = | x − a | ϕ (x), where ϕ (x) is continuous function, then

विकल्प

  • f' (a+) = ϕ (a)

  • f' (a) = −ϕ (a)

  • f' (a+) = f' (a)

  • none of these

MCQ

उत्तर

\[f'\left( a^+ \right) = \phi\left( a \right)\]

\[f'\left( a^- \right) = - \phi\left( a \right)\]

Here, 

\[f\left( x \right) = \left| x - a \right| \phi\left( x \right)\]
\[f'\left( a^+ \right) = \lim_{h \to 0} \frac{f\left( a + h \right) - f\left( a \right)}{h} = \lim_{h \to 0} \frac{\left| h + a - a \right|\phi\left( a + h \right) - \left| a - a \right|\phi\left( a \right)}{h} = \lim_{h \to 0} \frac{h \phi\left( a + h \right)}{h} = \lim_{h \to 0} \phi\left( a + h \right) = \phi\left( a \right)\]

Also,

\[f'\left( a^- \right) = \lim_{h \to 0} \frac{f\left( a - h \right) - f\left( a \right)}{h} = \lim_{h \to 0} \frac{\left| a - h - a \right| \phi\left( a - h \right) - \left| a - a \right| \phi\left( a \right)}{h} = \lim_{h \to 0} \frac{\left| - h \right| \phi\left( a - h \right)}{h} = \lim_{h \to 0} - \phi\left( a - h \right) = - \phi\left( a \right)\]
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अध्याय 9: Continuity - Exercise 9.4 [पृष्ठ ४२]

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आरडी शर्मा Mathematics [English] Class 12
अध्याय 9 Continuity
Exercise 9.4 | Q 2 | पृष्ठ ४२

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