Step 1: Understand the function
The given function is f(x) = (log x) / x, where log denotes the natural logarithm
Step 2: Find the first derivative
To find local maxima, we first differentiate f(x) using the quotient rule:
f'(x) = [ (1/x) * x - log x * 1 ] / x² = (1 - log x) / x².
Step 3: Find critical points
Set the derivative equal to zero to find critical points:
(1 - log x) / x² = 0 implies 1 - log x = 0
=> log x = 1
=> x = e (where e ≈ 2.718)
Step 4: Determine the nature of critical point
To check if x = e is a maximum, examine the second derivative or test values around x = e.
For x slightly less than e, f'(x) > 0 (function increasing).
For x slightly greater than e, f'(x) < 0 (function decreasing).
Hence, x = e is a point of local maximum.
Step 5: Conclusion
The function f(x) attains a local maximum at x = e.
Final Answer: x = e
If \( x = a(0 - \sin \theta) \), \( y = a(1 + \cos \theta) \), find \[ \frac{dy}{dx}. \]
Balance Sheet of Atharv and Anmol as at 31st March, 2024
Liabilities | Amount (₹) | Assets | Amount (₹) |
---|---|---|---|
Capitals: | Fixed Assets | 14,00,000 | |
Atharv | 8,00,000 | Stock | 4,90,000 |
Anmol | 4,00,000 | Debtors | 5,60,000 |
General Reserve | 3,50,000 | Cash | 10,000 |
Creditors | 9,10,000 | ||
Total | 24,60,000 | Total | 24,60,000 |