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Chapter 1: Problem 39

The range of the function \(f:[0,1] \rightarrow R, f(x)=x^{3}-x^{2}+4 x+2 \sin ^{-1} x\) is (a) \([-\pi-2,0]\) (b) \([2,3]\) (c) \([0,4+\pi]\) (d) \((0,2+\pi]\)

Short Answer

Expert verified
The range of the function \(f:[0,1] \rightarrow R, f(x)=x^{3}-x^{2}+4 x+2 \sin ^{-1} x\) is \([0, 4+\pi]\). So, the correct option is (c).

Step by step solution

01

Analyze the components of the function

The given function \(f(x)=x^{3}-x^{2}+4 x+2 \sin ^{-1} x\) can be broken down into two parts: \(g(x)=x^{3}-x^{2}+4 x\) which is a cubic polynomial and \(h(x)=2 \sin ^{-1} x\) which is a transformed inverse sine function. The range of the function will depend on how these two functions \(g(x)\) and \(h(x)\) behave within the domain [0,1].
02

Identify the endpoints

A function's positive or negative extremes can often occur at the endpoints of its domain, so we should evaluate \(f(x)\) at x=0 and x=1. Here, \(f(0)=0^3-0^2+4*0+2*\sin^{-1}(0)=0\) and \(f(1)=1^3-1^2+4*1+2*\sin^{-1}(1)=1-1+4+2*\frac{\pi}{2}=4+\pi\). Hence, the minimum value is 0 and the maximum value is \(4+\pi\).
03

Check for local extrema within the domain [0,1]

To find the local extrema, we need to take the derivative of the function, set it to zero and solve for x. The derivative of the function is \(f'(x)=3x^2-2x+4+2*\frac{1}{\sqrt{1-x^2}}\). Setting this equal to zero gives a complex equation. However, the derivative is undefined outside the domain [-1,1] for the inverse sine part, but our given domain is within [-1,1]. So, for x in [0,1], the derivative exist, and the function is increasing in this interval since the derivative is positive (all terms are positive for x in [0,1]). Hence, the function will not have any local minima or maxima in the interval (0,1).
04

Determine the range

With no local extrema in the domain (0,1) and the values at the endpoints determined, we can now conclude that the range of the function within the prescribed domain [0, 1] will be [the value of the function at 0, the value of the function at 1], which is [0, \(4+\pi\)].

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