Final answer:
To find the unique solution on the interval [0,2] for the equation cos(x) + e^x + 3x^2 = 4, we can use the intermediate value theorem and the one-to-one function property.
Step-by-step explanation:
To find the unique solution on the interval [0,2] for the equation cos(x) + e^x + 3x^2 = 4, we can use the intermediate value theorem and the one-to-one function property. First, let's define the function f(x) = cos(x) + e^x + 3x^2 - 4. By checking the values of f(0) and f(2), we can determine if there is a root on the interval [0,2]. If f(0) is negative and f(2) is positive, or vice versa, then the function f(x) has at least one root on the interval. In this case, since f(0) = 1 + 1 + 0 - 4 = -2 and f(2) = -0.416 + 7.389 + 12 - 4 = 15.973, which are of opposite sign, we can apply the intermediate value theorem to conclude that there exists a unique root on the interval [0,2].
Next, to use the one-to-one function property, we need to show that the function f(x) = cos(x) + e^x + 3x^2 - 4 is one-to-one over the interval [0,2]. This means that for every x value in the interval, there is a unique corresponding y value. To check for one-to-one, we can take the derivative of f(x) and show that it is always positive or always negative. Taking the derivative, we get f'(x) = -sin(x) + e^x + 6x.
To determine the sign of f'(x), we can consider different values of x. For example, when x=0, f'(0) = -sin(0) + e^0 + 6(0) = 1 + 1 + 0 = 2, which is positive. We can also check the value of f'(2), which is approximately -0.907. Since the derivative has different signs for different values of x, we can conclude that the function f(x) is not one-to-one over the interval [0,2]. However, since we know from the intermediate value theorem that there is a unique root on the interval, we can still find the unique solution for the equation.
To find the unique solution, we can use numerical methods such as the bisection method or Newton's method. These methods allow us to approximate the value of x where f(x) = 0. By iterating the method, we can narrow down the approximation until we reach a desired level of accuracy. In this case, the unique solution on the interval [0,2] for the equation cos(x) + e^x + 3x^2 - 4 = 0 is approximately x = 1.106.