query: If maximum value of $\int_0^1 (f(x))^3dx$ under the condition $-1\le f(x) \le 1$;$\int_0^1 f(x).dx=0$ is $\frac pq$, find $p+q$
positive_passages.docid: 88580
positive_passages.text: Finding the maximum value of $\int_0^1 f^3(x)dx$
negative_passages.docid: 106868
negative_passages.text: Prove that $x+\frac1{4x} ≥ 1$ for $x>0$
negative_passages.docid: 31503
negative_passages.text: Question about Maximum Modulus Principle applied to $|f|+|g|$
negative_passages.docid: 133851
negative_passages.text: How to arrive at $\int_{0}^{1} f^3(x) \, dx < (\int_{0}^{1} f(x) \, dx)^2$?
negative_passages.docid: 102943
negative_passages.text: $f(x)=1/q$ for $x=p/q$ is integrable
negative_passages.docid: 37015
negative_passages.text: Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$
negative_passages.docid: 23142
negative_passages.text: Suppose $x, y, z$ are positive real number such that $x + 2y + 3z = 1$. Find the maximum value of $xyz^2$
negative_passages.docid: 3273
negative_passages.text: Trouble in proving that $\|x\|_p = \max|x_j|$
negative_passages.docid: 144942
negative_passages.text: Obtain the value of $\int_0^1f(x) \ dx$, where $f(x) = \begin{cases} \frac {1}{n}, & \frac{1}{n+1}<x\le\frac{1}{n}\\ 0, & x=0 \end{cases}$
negative_passages.docid: 154945
negative_passages.text: $|px^2+qx+r|\le1$ for all $x$ in $[-1; 1]$, show that $|rx^2+qx+p|\le2$ for all $x$ in $[-1; 1]$
negative_passages.docid: 33097
negative_passages.text: Proving an inequality without an integral: $\frac {1}{x+1}\leq \ln (1+x)- \ln (x) \leq \frac {1}{x}$
negative_passages.docid: 151400
negative_passages.text: Closed-form of $\int_0^1 \operatorname{Li}_p(x) \, dx$
negative_passages.docid: 27705
negative_passages.text: Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
negative_passages.docid: 41619
negative_passages.text: How find the maximum value of $|bc|$
negative_passages.docid: 156599
negative_passages.text: Wirtinger Inequality: $\int_0^1 f'^2 \geq \pi^2 \int_0^1 f^2 $
negative_passages.docid: 85535
negative_passages.text: Finding maximum of $|f'(i)|$
negative_passages.docid: 34888
negative_passages.text: An MCQ for finding the extremal of the functional $J = \int_{a}^{b} F(x, y, y^{'})$