Let a, b ∈R be such that the fucntion f given by f(x)= ln|x|+b x^2+a x, x ≠0 has extreme values

step 1 In the equation, we are given a function fx, that is l n x plus bx square plus a x, where x is n

Let a, b ∈R be such that the fucntion f given by f(x)=...

Let $a, b \in R$ be such that the fucntion $f$ given by $f(x)=$ $\ln |x|+b x^{2}+a x, x \neq 0$ has extreme values at $x=-1$ and $x=2$. Statement $1: f$ has local maximum at $x=-1$ and at $x=2$ Statement $\mathbf{2}: a=\frac{1}{2}$ and $b=\frac{-1}{4}$. (A) Statement- 1 is false, Statement- 2 is true. (B) Statement- 1 is true, statement- 2 is true, statement-2 is a correct explanation for Statement-1. (C) Statement- $I$ is true, statement- 2 is true; statement- 2 is not a correct explanation for Statement- $1 .$ (D) Statement- 1 is true, statement- 2 is false.

Let $a, b \in R$ be such that the fucntion $f$ given by $f(x)=$ $\ln |x|+b x^{2}+a x, x \neq 0$ has extreme values at $x=-1$ and $x=2$.

Statement $1: f$ has local maximum at $x=-1$ and at $x=2$
Statement $\mathbf{2}: a=\frac{1}{2}$ and $b=\frac{-1}{4}$.
(A) Statement- 1 is false, Statement- 2 is true.
(B) Statement- 1 is true, statement- 2 is true, statement-2 is a correct explanation for Statement-1.
(C) Statement- $I$ is true, statement- 2 is true; statement- 2 is not a correct explanation for Statement- $1 .$
(D) Statement- 1 is true, statement- 2 is false.

Key Concepts

Extreme Values

Extreme values of a function refer to points where the function reaches either a local maximum or a local minimum. These are critical in understanding the behavior of a function, particularly in optimization problems. They occur where the first derivative is zero or undefined, serving as candidates for such extrema.

Critical Points and the First Derivative Test

Critical points are the values in the function's domain where its first derivative is zero or fails to exist. Determining these points is essential, as they are potential locations of local maxima or minima. The first derivative test is used to analyze the sign change of the derivative before and after a critical point, which helps in classifying the nature of the extremum.

Parameter Determination in Functional Equations

When functions contain parameters, these parameters can be found by applying given conditions, such as specific locations of extreme values or other functional behaviors. By substituting these conditions into the derivative or the function itself, one obtains equations that allow for the determination of the unknown parameters.

Second Derivative Test

The second derivative test further classifies the nature of a critical point by evaluating the concavity of the function at that point. If the second derivative is positive, the function is concave up and the critical point is a local minimum; if it is negative, the function is concave down and the critical point is a local maximum. This test provides a straightforward method to confirm the type of extremum, assuming the second derivative exists.

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