If a1 is the value of a for which function f(x)=x2+(a/x) has a local minimum at x=2 and a2 is the value of a for which f(x) has a point of inflection at x=1, then ((a1+a2/3)) is equal to

Question

If a1 is the value of a for which function f(x)=x2+(a/x) has a local minimum at x=2 and a2 is the value of a for which f(x) has a point of inflection at x=1, then ((a1+a2/3)) is equal to (A) 5 (B) 4 (C) 3 (D) 2

Tardigrade Admin · Accepted Answer

f prime(x)=2 x-(a/x2) f prime(2)=4-(a/4)=0 ⇒ a=16 ⇒ a1=16 Also, f prime prime( x )=2+(2 a / x 3) ∴ f prime prime(1)=2+2 a =0 ⇒ a =-1 ⇒ a 2=-1 Hence (( a 1+ a 2/3))=(15/3)=5

Q. 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 $(\frac{a _{1} + a _{2}}{3})$ is equal to

5

4

3

2

$f^{'} (x) = 2 x - \frac{a}{x ^{2}}$
$f^{'} (2) = 4 - \frac{a}{4} = 0 \Rightarrow a = 16 \Rightarrow a_{1} = 16$
Also, $f^{''} (x) = 2 + \frac{2 a}{x ^{3}}$
$∴ f^{''} (1) = 2 + 2 a = 0 \Rightarrow a = - 1 \Rightarrow a_{2} = - 1$
Hence $(\frac{a _{1} + a _{2}}{3}) = \frac{15}{3} = 5$

Q. If a1​ is the value of a for which function f(x)=x2+xa​ has a local minimum at x=2 and a2​ is the value of a for which f(x) has a point of inflection at x=1, then (3a1​+a2​​) is equal to

5

4

3

2

f′(x)=2x−x2a​ f′(2)=4−4a​=0⇒a=16⇒a1​=16 Also, f′′(x)=2+x32a​ ∴f′′(1)=2+2a=0⇒a=−1⇒a2​=−1 Hence (3a1​+a2​​)=315​=5

Q. 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 $(\frac{a _{1} + a _{2}}{3})$ is equal to

$f^{'} (x) = 2 x - \frac{a}{x ^{2}}$
$f^{'} (2) = 4 - \frac{a}{4} = 0 \Rightarrow a = 16 \Rightarrow a_{1} = 16$
Also, $f^{''} (x) = 2 + \frac{2 a}{x ^{3}}$
$∴ f^{''} (1) = 2 + 2 a = 0 \Rightarrow a = - 1 \Rightarrow a_{2} = - 1$
Hence $(\frac{a _{1} + a _{2}}{3}) = \frac{15}{3} = 5$