Let f(x)={ x^3+x^2-10 x, -1 ≤ x<0 cos x, 0 ≤ x<π / 2 1+sin x, π / 2 ≤

step 1 So in this question we have f of x is equal to x cube plus x square minus 10 x and x belongs to

Let f(x)={ x^3+x^2-10 x, -1 ≤ x<0 cos x, 0...

Let $f(x)=\left\{\begin{array}{lr}x^{3}+x^{2}-10 x, & -1 \leq x<0 \\ \cos x, & 0 \leq x<\pi / 2 \\ 1+\sin x, & \pi / 2 \leq x \leq \pi\end{array}\right.$ Then $\mathrm{f}(\mathrm{x})$ has (A) a local minimum at $x=\pi / 2$ (B) a global maximum at $\mathrm{x}=-1$ (C) an absolute minimum at $\mathrm{x}=-1$ (D) an absolute maximum at $\mathrm{x}=\pi$

Let $f(x)=\left\{\begin{array}{lr}x^{3}+x^{2}-10 x, & -1 \leq x<0 \\ \cos x, & 0 \leq x<\pi / 2 \\ 1+\sin x, & \pi / 2 \leq x \leq \pi\end{array}\right.$ Then $\mathrm{f}(\mathrm{x})$ has
(A) a local minimum at $x=\pi / 2$
(B) a global maximum at $\mathrm{x}=-1$
(C) an absolute minimum at $\mathrm{x}=-1$
(D) an absolute maximum at $\mathrm{x}=\pi$

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Key Concepts

Piecewise Functions

Functions defined by different formulas over separate intervals of their domain. This concept requires examining each interval individually for properties like continuity, differentiability, and behavior at the boundaries, which is crucial when determining extrema.

Local vs Global Extrema

Local extrema are points where the function attains a relative minimum or maximum within a small neighborhood, while global extrema (absolute extrema) are the highest or lowest values over the entire domain. Distinguishing between these is important when evaluating a function's overall extreme values.

Endpoint and Critical Point Analysis

When analyzing functions, especially piecewise-defined ones, it is essential to consider both critical points (where the derivative is zero or undefined) and endpoints of the intervals. Endpoints can be candidates for global extrema, and a careful analysis ensures that no potential maximum or minimum is overlooked.

Continuity and Differentiability

Understanding whether a function is continuous and differentiable across its entire domain, particularly at the junctions of different pieces, is vital. Discontinuities or non-differentiable points can significantly affect the behavior of the function and the existence of local or global extrema.

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