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Chapter 15: Problem 9

The function \(y(x)\) is given by \(y(x)=1-\cos x\). Find the values of \(x\) where (a) \(y^{\prime}=0\), (b) \(y^{\prime \prime}=0\).

Short Answer

Expert verified
Question: For the function \(y(x)=1-\cos x\), find the values of \(x\) where the first derivative and the second derivative are zero. Answer: (a) The first derivative is zero at \(x=n\pi\), where \(n\) is an integer. (b) The second derivative is zero at \(x=(2n+1)\frac{\pi}{2}\), where \(n\) is an integer.

Step by step solution

01

Compute the first derivative of y(x)

To find the first derivative, we will use the basic derivative rules for trigonometric functions: \((\cos x)'=-\sin x\). Differentiating the given function \(y(x)=1-\cos x\) with respect to \(x\), we get the first derivative: \(y^{\prime}(x)=-(-\sin x)=\sin x\).
02

Solve for x where the first derivative equals 0

Now, we need to find the values of \(x\) for which \(y^{\prime}(x)=\sin x=0\). This equation is satisfied whenever \(x=n\pi\), where \(n\) is an integer (\(n\in\mathbb{Z}\)). Thus, the values for x in this case are: \(x=n\pi\), where \(n\in\mathbb{Z}\).
03

Compute the second derivative of y(x)

Next, we need to compute the second derivative of \(y(x)=1-\cos x\). To compute the second derivative, we differentiate the first derivative, \(y^{\prime}(x)=\sin x\), with respect to \(x\). Using the basic derivative rules for trigonometric functions, we have: \((\sin x)'=\cos x\). Therefore, the second derivative of \(y(x)\) is: \(y^{\prime\prime}(x)=\cos x\).
04

Solve for x where the second derivative equals 0

Lastly, we want to find the values of \(x\) for which \(y^{\prime\prime}(x)=\cos x=0\). This equation is satisfied when \(x=(2n+1)\frac{\pi}{2}\), where \(n\) is an integer (\(n\in \mathbb{Z}\)). Thus, the values for x in this case are: \(x=(2n+1)\frac{\pi}{2}\), where \(n\in\mathbb{Z}\). To sum up the findings: (a) The first derivative is zero at \(x=n\pi\), where \(n\) is an integer, and (b) The second derivative is zero at \(x=(2n+1)\frac{\pi}{2}\), where \(n\) is an integer.

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