Integration inequality

Whether ∫ba‖f(x)‖2dx≤sup{‖f(x)‖2:x∈[a,b]}(b−a)\int_{a}^{b} \|f(x)\|^2 dx \leq \sup \{ \|f(x)\|^2: x \in [a, b]\}(b-a) is true? Here \|\cdot \|\|\cdot \| denotes the Euclidean norm and f:[a, b]\rightarrow \mathbb{R}^nf:[a, b]\rightarrow \mathbb{R}^n so that f(x)=(f_1(x), f_2(x),…,f_n(x)),f(x)=(f_1(x), f_2(x),…,f_n(x)), i. e ff is an n-real vector function on [a, b][a, b]

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Hint: Rewrite the right-hand side as \int_a^b \|f\|_\infty^2 \,dx\int_a^b \|f\|_\infty^2 \,dx and use the positivity of the integral.
– MaoWao
2 days ago

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