Question

9. [3?] Suppose that $f: [0,1] \rightarrow \mathbb{R}$ be continuous. Let $M = \max\{ |f(x)| : x \in [0,1] \}$. Prove that \begin{equation} \lim_{n \rightarrow \infty} \left( \int_0^1 |f(x)|^n dx \right)^{1/n} = M. \end{equation}

          9. [3?] Suppose that $f: [0,1] \rightarrow \mathbb{R}$ be continuous. Let $M = \max\{ |f(x)| : x \in [0,1] \}$. Prove that \begin{equation*} \lim_{n \rightarrow \infty} \left( \int_0^1 |f(x)|^n dx \right)^{1/n} = M. \end{equation*}

$9. [3?] Suppose that f: [0,1] →ℝ be continuous. Let M = max{ |f(x)| : x ∈ [0,1] }. Prove that limn →∞( ∫0^1 |f(x)|^n dx )^1/n = M.$

Added by Nicole V.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Nicole Smina

Step 1

Let M = max{|f(x)| : x ∈ [0,1]}. Then M ≥ 0. If M = 0, then |f| ≡ 0, so ∫_0^1 |f(x)|^n dx = 0 for all n and the nth root is 0; the limit equals M. So assume M > 0. Show more…

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