91Ó°ÊÓ

The first piece of the function is \(f(x) = x^3 + x^2 - 10x\) for \(-1 \leq x < 0\). To find the local extrema, we need to find the critical points where the derivative is zero or undefined. The derivative for this piece is \(f'(x) = 3x^2 + 2x - 10\). Setting this equal to zero, we have: \(3x^2 + 2x - 10 = 0\) However, there is no real solution for \(x\) in the range of \(-1 \leq x < 0\). So, no local extrema exist for this piece of the function. Check endpoints of the interval: \(f(-1) = (-1)^3 + (-1)^2 - 10(-1) = -1 + 1 + 10 = 10\) There is a value at \(x = -1\).

The second piece of the function is \(f(x) = \cos x\) for \(0 \leq x < \pi/2\). The derivative for this piece is \(f'(x) = -\sin x\). Since \(\sin\) function ranges between \(-1\) and \(1\), the derivative will not be undefined. The local extrema occur when \(-\sin x = 0 \Rightarrow x = 0\). However, this is an endpoint of the interval and must be analyzed along with the other interval endpoints. Check endpoints of the interval: \(f(0) = \cos 0 = 1\)

The third piece of the function is \(f(x) = 1 + \sin x\) for \(\pi/2 \leq x \leq \pi\). The derivative for this piece is \(f'(x) = \cos x\). As \(\cos x\) ranges between \(-1\) and \(1\), the local extrema occur when \(\cos x = 0 \Rightarrow x = \pi/2\). Check endpoints of the interval: \(f(\pi/2) = 1 + \sin(\pi/2) = 1 + 1 = 2\) \(f(\pi) = 1 + \sin(\pi) = 1\)

From the previous steps, we have the following extrema values: \(f(-1) = 10\) \(f(0) = 1\) \(f(\pi/2) = 2\) \(f(\pi) = 1\) From the given options, we now analyze their validity: (A) Local minimum at \(x = \pi/2\): Since \(f(\pi/2) = 2\) and \(f(-1) = 10\), this is not a local minimum, so this option is incorrect. (B) Global maximum at \(x = -1\): At \(x = -1\), we have a function value of \(10\), which is the highest among the extrema values, so this option is correct. (C) Absolute minimum at \(x = -1\): Considering \(x=0\), the function value is \(1\), which is lower than the value at \(x=-1\). So, this option is incorrect. (D) Absolute maximum at \(x = \pi\): At \(x = \pi\), the function value is \(1\), which is not the maximum, so this option is incorrect. Thus, the correct answer is (B) a global maximum at \(x = -1\).