By the mean value theorem, the function y = sin x² on the interval [0, π/4] has this guaranteed number: x ≈ 0.460.
How to find a number that guarantees the mean value theorem
According to the mean value theorem, a function that is differentiable at an interval [a, b] has a number c within the interval such that:
f'(c) = [f(b) - f(a)]/(b - a) (1)
The first derivative of the function is f'(x) = 2 · x · cos x² and by the mean value theorem we know that:
2 · c · cos c² = [sin (π/4)²- sin 0²]/(π/4 - 0)
2 · c · cos c² = [- √2 / 2]/(π /4)
2 · c · cos c² ≈ 0.900
This equation cannot be solved analytically. By the resource of a graphing tool we found the following solution within the interval: c ≈ 0.460.
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