Upper Bounds 4 Assume That F Is Twice Differentiable On 0 And Mo M M Are The Finite Least Of F X F X F 1 (153.28 KiB) Viewed 36 times
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upper bounds 4. Assume that f is twice differentiable on (0, ∞), and Mo, M₁, M₂ are the finite least of f(x)\, f'(x). [f"(r)], respectively, on (0.00). Show that M² 4M,M₂. Hint: For x > 0 and h > 0. Taylor's Theorem says that f(x + h) = f(x) + f'(x)h + fƒ"(c)h²/2, for 2Mo h some c in (x,x+h). Thus f'(x) = (f(x+h) − f(x))/h - hf"(c). It follows that M₁ < h 2Mo h Find the infimum of M₂. + M₂. h 2
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