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

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03:55

Frank Lin

Calculus 1 / AB

Chapter 5

Integrals

Section 3

The Fundamental Theorem of Calculus

Integration

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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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

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