Question

Show that f(x)=sin(x ²+3x) is equal to g(x)=−xcos(x) at least once on [−−,− Hint: Think about the IVT.

Answer 1

To show that f(x) = sin(x^2 + 3x) is equal to g(x) = -xcos(x) at least once on the interval [-∞,∞], we can use the Intermediate Value Theorem (IVT).

Step-by-step explanation:

To show that f(x) = sin(x^2 + 3x) is equal to g(x) = -xcos(x) at least once on the interval [-∞,∞], we can use the Intermediate Value Theorem (IVT).

The IVT states that if a function is continuous on a closed interval [a, b] and takes on two different values, say c and d, then it must take on every value between c and d at least once.

In this case, we can see that both functions are continuous on the entire real number line and they both take on values between -1 and 1. Therefore, there must exist at least one value of x for which f(x) = g(x).