If x=-1 and x=2 are extreme points of f(x)=αlog|x|+βx^2+x, then(A) α=-6, β=(1)/(2) (B) α=-6, β=-

step 1 Here f x is given, so we can write f -dash x equal alpha by x plus 2 beta x plus 1. So equation

If x=-1 and x=2 are extreme points of f(x)=αlog|x|+βx^2+x,...

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \log |x|+\beta x^{2}+x$, then(A) $\alpha=-6, \beta=\frac{1}{2}$
(B) $\alpha=-6, \beta=-\frac{1}{2}$
(C) $\alpha=2, \beta=-\frac{1}{2}$
(D) $\alpha=2, \beta=\frac{1}{2}$

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Key Concepts

Solving Systems of Equations

When parameters within a function are unknown, imposing conditions like setting the derivative equal to zero at given critical points yields a system of equations. Solving these simultaneous equations is a fundamental algebraic process that allows one to determine the values of the unknown parameters that satisfy the conditions imposed on the function.

Critical Points

Critical points are values of the independent variable where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or saddle points. In calculus, finding these points is essential in the analysis of the behavior of functions, especially when determining where a function reaches extreme values.

Differentiation of Logarithmic Functions

Differentiating logarithmic functions, particularly those involving the absolute value such as log|x|, involves applying the chain rule and recognizing that the derivative of log|x| is 1/x for x ? 0. This concept is crucial when working with functions that include logarithmic components, as it allows one to correctly compute derivatives and analyze function behavior.

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