The function f(x)=(4 sin^2 x-1)^n(e^x-x+1) n ∈N, has a local minimum at x=(π)/(6), then (A) n ca

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The function f(x)=(4 sin^2 x-1)^n(e^x-x+1) n ∈N, has a local...

The function $f(x)=\left(4 \sin ^{2} x-1\right)^{n}\left(e^{x}-x+1\right)$ $\mathrm{n} \in \mathrm{N}$, has a local minimum at $\mathrm{x}=\frac{\pi}{6}$, then (A) n can be any even natural number (B) $\mathrm{n}$ can be an odd natural number (C) $\mathrm{n}$ can be odd prime number (D) n can be any natural number

The function $f(x)=\left(4 \sin ^{2} x-1\right)^{n}\left(e^{x}-x+1\right)$
$\mathrm{n} \in \mathrm{N}$, has a local minimum at $\mathrm{x}=\frac{\pi}{6}$, then
(A) n can be any even natural number
(B) $\mathrm{n}$ can be an odd natural number
(C) $\mathrm{n}$ can be odd prime number
(D) n can be any natural number

Key Concepts

Local Extrema

A local extremum (minimum or maximum) of a function is a point where its value is less than or equal to (or greater than or equal to) the values at nearby points. Typically, these are identified by checking that the first derivative is zero (or undefined) and that the second derivative test (or other criteria) confirms the nature of the extremum.

Even and Odd Exponents

The difference between even and odd exponents is crucial when analyzing the behavior of a function near a point where one of its factors is zero. An even exponent tends to 'flatten' the graph near the zero, making the powered term non-negative irrespective of the sign of the base, while an odd exponent preserves the sign of the base and may create a sign change that affects the local extrema properties.

Multiplicity of Zeros

The concept of multiplicity of zeros indicates how many times a root occurs when a function is factored. A higher multiplicity, especially even multiplicity, means the graph touches the axis without crossing it, which is important for determining whether the function has a local minimum or maximum at that point.

Parameter Influence on Function Behavior

Parameters, such as the exponent in a function, can significantly influence its overall behavior, including the existence and nature of local extrema. By adjusting these parameters, one can ensure that certain desirable conditions, like having a local minimum at a particular point, are met, often by affecting the 'flatness' or curvature of the function near the point.

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