Given P(x)=x^4+a x^3+b x^2+c x+d such that x=0 is the only...

Given $P(x)=x^{4}+a x^{3}+b x^{2}+c x+d$ such that $x=0$ is the only real root of $P^{\prime}(x)=0$. If $P(-1)<P(1)$, then in the interval $[-1,1]$ $[2009$ (A) $P(-1)$ is the minimum and $P(1)$ is the maximum of $\mathrm{P}$ (B) $P(-1)$ is not minimum but $P(1)$ is the maximum of $\bar{P}$ (C) $P(-1)$ is the minimum and $P(1)$ is not the maximum of $P$ (D) neither $P(-1)$ is the minimum nor $P(1)$ is the maximum of $P$

Given $P(x)=x^{4}+a x^{3}+b x^{2}+c x+d$ such that $x=0$ is the only real root of $P^{\prime}(x)=0$. If $P(-1)<P(1)$, then in the interval $[-1,1]$ $[2009$
(A) $P(-1)$ is the minimum and $P(1)$ is the maximum of $\mathrm{P}$
(B) $P(-1)$ is not minimum but $P(1)$ is the maximum of $\bar{P}$
(C) $P(-1)$ is the minimum and $P(1)$ is not the maximum of $P$
(D) neither $P(-1)$ is the minimum nor $P(1)$ is the maximum of $P$

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Key Concepts

Polynomial Functions

A polynomial function is an expression involving sums of powers of a variable multiplied by constants. The overall degree and the coefficients dictate its behavior, including the number and nature of turning points, end behavior, and continuity properties.

Critical Points and Derivatives

Critical points are the values of the variable where the derivative of a function is zero or undefined. They indicate where the function's slope changes, which possibly correspond to local minima, maxima, or points of inflection, making them vital for understanding the function’s behavior.

Extreme Value Theorem

The Extreme Value Theorem states that any continuous function on a closed interval attains both a minimum and a maximum value somewhere within that interval. This theorem underpins optimization over closed intervals by ensuring that an optimum exists.

Monotonicity and Endpoint Analysis

The sign of a function's derivative across an interval provides insight into its monotonic behavior—whether it is consistently increasing or decreasing—which, in turn, aids in deciding if the extrema occur at interior critical points or at the interval endpoints.

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