If \( \alpha \) and \( \beta \) are absolute maximum and absolute minimum values of a function \( f(x)=\displaystyle\frac{x}{2}+\frac{2}{x} \) on the interval \( [1,2] \), then the value of \( \alpha+\beta \) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 9/2
On a closed interval the absolute extrema occur at a critical point or at an end point.
\( \displaystyle f'(x)=\frac{1}{2}-\frac{2}{x^{2}} \)
\( \displaystyle =0 \)
\( \displaystyle x^{2}=4 \)
The only critical point in range is x = 2, which is itself an end point. So only the two end values need comparing.
\( \displaystyle f(1)=\frac{1}{2}+2=\frac{5}{2} \)
\( \displaystyle f(2)=1+1=2 \)
The absolute maximum is \( \alpha=\frac{5}{2} \) and the absolute minimum is \( \beta=2 \), so
\( \displaystyle \alpha+\beta=\frac{9}{2} \)