Question

1. show that there is a value c with 0 < c < 2 such that x² + cos πx = 4. then, use a graphing utility to find the approximate value of c.

Explanation:

Step1: Define a function

Let \(f(x)=x^{2}+\cos(\pi x)-4\). This function is continuous on the interval \([0,2]\) since \(y = x^{2}\), \(y=\cos(\pi x)\) and \(y = - 4\) are all continuous functions, and the sum of continuous functions is continuous.

Step2: Evaluate the function at the endpoints

When \(x = 0\), \(f(0)=0^{2}+\cos(0)-4=1 - 4=-3\). When \(x = 2\), \(f(2)=2^{2}+\cos(2\pi)-4=4 + 1-4=1\).

Step3: Apply the Intermediate - Value Theorem

Since \(f(x)\) is continuous on \([0,2]\) and \(f(0)=-3<0\) and \(f(2)=1>0\), by the Intermediate - Value Theorem, which states that if \(y = f(x)\) is continuous on a closed interval \([a,b]\) and \(k\) is a number between \(f(a)\) and \(f(b)\), then there exists at least one number \(c\) in the open interval \((a,b)\) such that \(f(c)=k\). Here, \(k = 0\), so there exists a \(c\in(0,2)\) such that \(f(c)=0\), which means \(c^{2}+\cos(\pi c)=4\).

Step4: Use a graphing utility

Using a graphing calculator or software (such as Desmos), graph \(y=x^{2}+\cos(\pi x)-4\) and find the \(x\) - intercept in the interval \((0,2)\). The approximate value of \(c\approx1.67\).

Answer:

The existence of \(c\) is shown by the Intermediate - Value Theorem and the approximate value of \(c\) is \(1.67\).