Question

If $a_1$ is the value of a for which function $f(x)=x^2+\frac{a}{x}$ has a local minimum at $x=2$ and $a_2$ is the value of a for which $f(x)$ has a point of inflection at $x=1$, then $\left(\frac{a_1+a_2}{3}\right)$ is equal to

• A. 5
• B. 4
• C. 3
• D. 2

Answer 1

Answer: (A)

$f^{\prime}(x)=2 x-\frac{a}{x^2}$
$f^{\prime}(2)=4-\frac{a}{4}=0 \Rightarrow a=16 \Rightarrow a_1=16$
Also, $f ^{\prime \prime}( x )=2+\frac{2 a }{ x ^3}$
$\therefore f ^{\prime \prime}(1)=2+2 a =0 \Rightarrow a =-1 \Rightarrow a _2=-1$
Hence $\left(\frac{ a _1+ a _2}{3}\right)=\frac{15}{3}=5$