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Let $I=\displaystyle\int_a^b(x^4-2x^2)\,dx$. Find the ordered pair $(a,b)$ that minimises $I$.
$x^4-2x^2=x^2(x^2-2)\leq0$ when $|x|\leq\sqrt{2}$ and $\geq0$ outside.
To minimise the integral (make it most negative), integrate over the region where the integrand is most negative — i.e., $[-\sqrt{2},\sqrt{2}]$ where the function is $\leq0$.
Any sub-interval of $[-\sqrt{2},\sqrt{2}]$ gives a more negative (or equal) value, and the widest such interval $[-\sqrt{2},\sqrt{2}]$ gives the minimum.
$(a,b)=(-\sqrt{2},\sqrt{2})$
The genetic information inherited from your parents for a specific trait like hair color is called your:
Genotype refers to the actual alleles present (e.g., BB, Bb, bb). Phenotype is the physical expression. Karyotype is the chromosome set.
The ester formed from a fatty acid and a long-chain alcohol is called:
Waxes are esters of long-chain fatty acids with long-chain alcohols. Acylglycerols (fats/oils) are esters with glycerol. Phospholipids have a phosphate group.
$t$ is a positive integer and $\dfrac{4}{7} = \dfrac{t}{s}$.
Compare:
Column A: $s$
Column B: $7$
From $\dfrac{4}{7} = \dfrac{t}{s}$, we get $s = \dfrac{7t}{4}$.
Since $t$ is a positive integer, different values of $t$ yield different results:
- If $t = 4$: $s = 7$ → Column A $=$ Column B (equal)
- If $t = 8$: $s = 14 > 7$ → Column A is greater
- If $t = 2$: $s = 3.5 < 7$ → Column B is greater
Since the result changes depending on $t$, the relationship cannot be determined from the information given.
Three adults—R, S, and V—will be traveling in a van with five children—F, H, J, L, and M. The van has a driver's seat and one passenger seat in the front, and two benches behind the front seats, one bench behind the first. Each bench has room for exactly three people. Everyone must sit in a seat or on a bench, and seating is subject to the following restrictions: An adult must sit on each bench. Either R or S must sit in the driver's seat. J must sit immediately beside M.
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