An even polynomal function f(x) satisfies a relation f(2 x)(1-f((1)/(2 x)))+f(16 x^2 y)=f(-2)-f(

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An even polynomal function f(x) satisfies a relation f(2...

An even polynomal function $f(x)$ satisfies a relation $f(2 x)\left(1-f\left(\frac{1}{2 x}\right)\right)+f\left(16 x^{2} y\right)=f(-2)-f(4 x y) \forall x, y \in$ $\mathrm{R}-\{0\}$ and $\mathrm{f}(4)=-255, \mathrm{f}(0)=1$ Which of the following holds good? (A) $\mathrm{f}(\mathrm{x})$ has local maximum at $\mathrm{x}=1$. (B) $f(x) f\left(\frac{1}{x}\right) \leq 0$ (C) Range of values of $\mathrm{k}$ for which $|\mathrm{f}(\mathrm{x})|=\mathrm{k}-2$ has exactly four distinct solutions is $(2,3)$. (D) $\int_{ }^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{3}{4}$.

An even polynomal function $f(x)$ satisfies a relation
$f(2 x)\left(1-f\left(\frac{1}{2 x}\right)\right)+f\left(16 x^{2} y\right)=f(-2)-f(4 x y) \forall x, y \in$
$\mathrm{R}-\{0\}$ and $\mathrm{f}(4)=-255, \mathrm{f}(0)=1$
Which of the following holds good?
(A) $\mathrm{f}(\mathrm{x})$ has local maximum at $\mathrm{x}=1$.
(B) $f(x) f\left(\frac{1}{x}\right) \leq 0$
(C) Range of values of $\mathrm{k}$ for which $|\mathrm{f}(\mathrm{x})|=\mathrm{k}-2$ has exactly four distinct solutions is $(2,3)$.
(D) $\int_{ }^{1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{3}{4}$.

Key Concepts

Definite Integrals of Polynomial Functions

The computation of definite integrals of polynomial functions uses the fundamental theorem of calculus, which involves finding an antiderivative of the function and evaluating it across the interval. This process is straightforward with polynomials due to their well-understood behavior and simple integration rules.

Even Functions

Even functions satisfy the property f(x) = f(-x) for all x in their domain, which means their graphs are symmetric with respect to the y-axis. In the context of polynomial functions, this often results in the function containing only even-powered terms, simplifying its analysis in symmetry and integral computations.

Functional Equations

Functional equations involve finding functions that satisfy a given equation in which the function appears at several arguments. Analyzing such equations often requires substitutions and exploiting symmetries to deduce properties of the function or its specific form.

Local Extrema in Calculus

Local extrema, such as local maxima and minima, are points in the domain of a function where the function attains a peak or a trough relative to nearby points. Determining a local maximum typically involves checking that the first derivative is zero and that the second derivative is negative at that point.

Inequalities Involving Function Products

Analyzing inequalities like f(x)·f(1/x) ? 0 involves studying the sign of the function across its domain. This requires an understanding of the function’s behavior, including its symmetry and the location of its zeros, to determine where the product of the function evaluated at reciprocal points is non-positive.

Solution Counting in Parameterized Equations

Determining the number of distinct solutions to equations that depend on a parameter involves analyzing how the roots of the equation change as the parameter varies. In doing so, one examines the structure of the equation, potential multiplicities, and the behavior of the function to establish intervals for the parameter that yield a specific number of solutions.

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