at least one
[at least one|at least one] The Mean Value Theorem states that for a function f(x) = vx over the interval [0, 9], there exists at least one value c ∈ (0, 9) where f'(c) equals the slope between (0, f(0)) and (9, f(9)).
Definition
at least one The Mean <a href='/en/entity/value-theorem'>Value Theorem</a> states that for a function f(x) = vx over the interval [0, 9], there exists at least one value c ∈ (0, 9) where f'(c) equals the slope between (0, f(0)) and (9, f(9)). Example 4.15 demonstrates verifying the theorem's applicability by showing the function satisfies its hypotheses. The theorem guarantees at least one solution exists within the interval, though it does not ensure a unique solution.
Mechanism
at least one The existence of at least one point c in the interval (a, b) is established by introducing a function that meets <a href='/en/entity/rolle-s-theorem'>Rolle's theorem</a> criteria. This function enables the application of Rolle's theorem, which leads to the conclusion that f(b) - f(a) equals the derivative at c. The proof relies on constructing an auxiliary function that satisfies the necessary conditions for Rolle's theorem to apply.
Causes
at least one The function f(x) = x - cosx has at least one zero, as demonstrated by Example 4.15. This example shows that the Mean <a href='/en/entity/value-theorem'>Value Theorem</a> applies to f(x) = vx over [0, 9], ensuring there exists at least one value c in (0, 9) where f'(c) equals the slope between (0, f(0)) and (9, f(9)). The theorem's conditions are verified through this example, confirming the existence of at least one solution.
Effects
at least one The function f(x) = x - cosx has at least one zero, as demonstrated by Example 4.15. This example shows that the Mean <a href='/en/entity/value-theorem'>Value Theorem</a> applies to f(x) = vx over [0, 9], ensuring there exists at least one value c in (0, 9) where f'(c) equals the slope between (0, f(0)) and (9, f(9)). The theorem's application confirms the existence of at least one solution for the equation x - cosx = 0. Both examples highlight how the Mean Value Theorem can be used to verify the existence of specific values satisfying certain conditions.