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Debora [2.8K]
3 years ago
8

Find the interval on which the curve of y equals the integral from 0 to x of 2 divided by the quantity 1 plus 3 times t plus t s

quared, dt is concave up.
Mathematics
1 answer:
elena-14-01-66 [18.8K]3 years ago
6 0

y=\displaystyle\int_0^x\frac2{1+3t+t^2}\,\mathrm dt

By the fundamental theorem of calculus,

\dfrac{\mathrm dy}{\mathrm dt}=\dfrac2{1+3x+x^2}

then by standard differentiation,

\dfrac{\mathrm d^2y}{\mathrm dt^2}=-\dfrac{6+4x}{(1+3x+x^2)^2}

y is concave upward whenever \dfrac{\mathrm d^2y}{\mathrm dt^2}>0. The denominator is always positive, so the sign is determined by the numerator alone:

6+4x>0\implies x>-\dfrac32

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