Correct Answer - Option 4 :
\(\frac{1}{2}\)
Concept:
Following steps to finding maxima and minima using derivatives.
Find the derivative of the function.
Set the derivative equal to 0 and solve. This gives the values of the maximum and minimum points.
Now we have to find the second derivative.
- f"(x) is less than 0 then the given function is said to be maxima
- If f"(x) Is greater than 0 then the function is said to be minima
\(\rm sin2 x =2sinx \ . cosx\)
Calculation:
Let, f(x) = \(\rm sinx \ . cosx\)
= \(\rm\frac12\times sin2 x \) (∵ \(\rm sin2 x =2sinx \ . cosx\))
f'(x) = \(\rm\frac12\times (2cos2 x)\)
= cos 2x
Now, f'(x) = 0 ⇒ cos 2x = 0
We know, \(\rm cos(\fracπ2)=0\)
∴ cos 2x = \(\rm cos(\fracπ2)\)
⇒ 2x = \(\fracπ2\)
⇒ x = \(\fracπ4\)
Also, f''(x) = \(\rm-(4sin2 x)\)
⇒ f''(\(\fracπ4\)) = - 4 < 0
At x = \(\fracπ4\) , f(x) is maximum.
∴ f(\(\fracπ4\)) = \(\rm\frac12\times sin (\frac{\pi}{2})\)
= \(\frac12\)
Hence, option (4) is correct.