Let the real values a, b be such that the function f given by f(x)=ln|x|+b x^2+a x, x ≠0 has ext

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

Let the real values a, b be such that the function f given...

Let the real values $a, b$ be such that the function $\mathrm{f}$ 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 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 1 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 the real values $a, b$ be such that the function $\mathrm{f}$ 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 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 1 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

Derivative and Critical Points

In calculus, determining local extrema involves finding the critical points where the first derivative of the function is zero or undefined. This concept is key because setting the derivative equal to zero yields the necessary conditions for potential local minima or maxima, which then need to be verified using additional tests.

Parameter Determination Through Simultaneous Equations

When a function is composed of parameters, imposing conditions such as having extrema at specific points allows one to set up a system of equations. Solving these simultaneous equations derived from the derivative's critical point condition helps find the precise values of the parameters that ensure the function exhibits the desired behavior.

Second Derivative Test for Classification of Extrema

Once the critical points are identified, the second derivative test is a common method for classifying these points. The sign of the second derivative at a critical point indicates whether it is a local minimum (if the second derivative is positive) or a local maximum (if the second derivative is negative).

Differentiation of Logarithmic Functions

The differentiation of logarithmic functions, particularly the natural logarithm, is a fundamental concept in calculus. The derivative of ln|x| is 1/x for all x ? 0, and understanding this differentiation rule is crucial when the logarithm is part of a larger function whose extrema are being investigated.

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