Show that the following equations have at least one solution in the given intervals. a. a cosa 2a2 + 3a 1 = 0, [0.2, 0.31 and [1.2, 1.3] b. (a-2)2 -InA =0, [1,2] and [e, 4] c. 2a cos(2a) -(a - 2)2 = 0. [2, 3] and [3,4] d. a (In A)r = 0, [4,5]

Problem 1

Show that the following equations have at least one solution in the given intervals.

a. x cos x- 2x2 + 3x -1 = 0, [0.2, 0.3] and [1.2, 1.3]

b. (x-2)2 -In x =0, [1,2] and [e, 4]

c. 2x cos(2x) -(x - 2)2 = 0. [2, 3] and [3,4]

d. x- (In x)x = 0, [4,5].

Step-by-step solution

Step 1 of 16

(a)

(i) Consider the function defined by

The function we will consider when discussing numerical methods will be assumed to be continuous because there is a minimal requirement for predictable behavior.

However, every polynomial function, exponential function, sine function, cosine function and constant function are continuous everywhere.

Therefore, is continuous on.

In addition

Now,