11. show that there is a value c with 0 < c < 2 such that x² + cos πx = 4. then, use a graphing utility to…

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).