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Problem 65 Hard Difficulty

On what interval is the curve $$ y = \displaystyle \int^x_0 \frac{t^2}{t^2 + t + 2} \, dt $$ concave downward?

Answer

$$
(-4,0)
$$

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Video Transcript

we know that why is crank it down on the specific intervals where why prime are the derivative is decreasing, So let's calculate the derivative first. Now we need to figure out where y double prime. In other words, the second derivative is less than zero to figure out where wide prime the first derivative is decreasing number off one G myself, you won over G squared. You can use the quotient rule because we have a numerator and denominator over here to figure this out. Zero equal to X squared plus four ax. We get access. Negative floor comma zero X squared plus works is upward opening problem. The word sister problem that opens upward there for the integral neck. The interval negative four comma zero is negative. So, in other words, negative for commas. Here is where it's con caved ounce. That's our solution