If [tex]$f(x)=a x^3+\frac{b}{x}+5$[/tex] has local minimum at [tex]$(2,-3)$[/tex], then what are the values of [tex]$a$[/tex] and [tex]$b$[/tex]?
Answer (2)
The values of a and b for the function f ( x ) = a x 3 + x b + 5 that has a local minimum at ( 2 , − 3 ) are a = − 4 1 and b = − 12 .
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To solve for the values of a and b given that the function f ( x ) = a x 3 + x b + 5 has a local minimum at the point ( 2 , − 3 ) , we have to use the conditions for local extrema in calculus.
Step 1: Use the condition of the function value.
Since the function has a local minimum at ( 2 , − 3 ) , then: f ( 2 ) = − 3 Substituting x = 2 in the function, we have: f ( 2 ) = a ( 2 ) 3 + 2 b + 5 = − 3 Simplifying this gives: 8 a + 2 b + 5 = − 3 8 a + 2 b = − 8 (Equation 1)
Step 2: Differentiate the function.
The first derivative of f ( x ) with respect to x is: f ′ ( x ) = 3 a x 2 − x 2 b At a local minimum, the derivative equals zero at x = 2 :
f ′ ( 2 ) = 3 a ( 2 ) 2 − 2 2 b = 0 12 a − 4 b = 0 12 a = 4 b Multiply through by 4 to clear the fraction: 48 a = b (Equation 2)
Step 3: Solve the system of equations.
From Equation 2, substitute b = 48 a into Equation 1: 8 a + 2 48 a = − 8 8 a + 24 a = − 8 32 a = − 8 a = − 4 1
Using a = − 4 1 in Equation 2, compute b :
b = 48 ( − 4 1 ) b = − 12
Conclusion
The values of a and b that satisfy the given conditions are: a = − 4 1 , b = − 12
