Class 12 Applications Of Derivatives · Short Answer2 Marksdisconnected_domain_monotonicity

Class 12 Maths · Applications Of Derivatives · Short Answer #51

Is there a nonzero real b for which f(x)=b/x is strictly increasing on the entire ordered domain R∖{0}? Distinguish this from increase on each domain component.

✅ Step-by-Step Solution

Final Answer: False globally; b<0 gives strict increase on each component only.

f′=−b/x². If b>0 the function decreases on each component; b=0 is excluded.
If b<0 the derivative is positive on each component, but −1<1 while f(−1)=−b>b=f(1), violating global strict increase.
Thus no nonzero b works on the entire domain; every b<0 works separately on (−∞,0) and (0,∞).

📋 CBSE Stepwise Marking Scheme

[0.5 marks] f′=−b/x². If b>0 the function decreases on each component; b=0 is excluded.
[1 mark] If b<0 the derivative is positive on each component, but −1<1 while f(−1)=−b>b=f(1), violating global strict increase.
[0.5 marks] Thus no nonzero b works on the entire domain; every b<0 works separately on (−∞,0) and (0,∞).

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Class 12 Maths · Applications Of Derivatives · Short Answer #51

✅ Step-by-Step Solution

📋 CBSE Stepwise Marking Scheme

Master Chapter 6: Applications Of Derivatives

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