To determine the local minimum of the function
f(x)=‌+‌, we start by finding the derivative of the function. This will help identify the critical points where potential minima or maxima might occur.
The derivative of
f(x) is found using the power rule and the chain rule:
f′(x)=‌(‌)+‌(‌)=‌−‌.Set the derivative equal to zero to find the critical points:
‌−‌=0Rearranging, we get:
‌=‌Multiplying both sides by
x2 to clear the fraction:
2=‌x2Multiplying both sides by 2 :
4=x2‌. ‌Thus,
x=±2.
Now, we evaluate whether these critical points are minima or maxima using the second derivative test.
f′′(x)=‌(‌−‌)=0+‌(−‌)=‌‌. ‌At
x=2 :
f′′(2)=‌=‌=‌>0‌. ‌ This indicates that at
x=2, the function
f has a local minimum.
At
x=−2 :
f′′(−2)=‌=‌=−‌<0‌. ‌This indicates that at
x=−2, the function
f has a local maximum.
Option C and Option D (i.e., for
x=0 and
x=1 ) are not even candidates given by solving the critical point equation, and moreover,
f(x) is not defined for
x=0 due to division by zero.
Therefore, the function
f(x)=‌+‌ has a local minimum at
x=2.
So, the correct answer is:
Option A:
x=2