Answer: the interval on which the curve is concave up is (-ā, -1/2)======
we find the interval where this curve is concave up by finding the interval in which the 2nd derivative is positive.differentiate the curve equation with respect to x
let the function f denote be the anti-derivative of the inside expression, i.e. a function such that
.then use the fundamental theorem of calculus:
differentiate again to get second derivative. we can use chain rule with 1/x as the outside function and 1 + x + xĀ² as the inside function
find critical numbers of the second derivative. these numbers are where the 2nd deriv. is undefined or zero.where the 2nd derivative is zero: set (dĀ²y/dxĀ²) to equal 0 and solve for x
where the 2nd derivative is undefined: if the denominator is equal to zero, the derivative will be undefeined. attempting to solve the equation 1 + x + xĀ² = 0results in a discriminant of
meaning that the equal 1 + x + xĀ² = 0 has no real solutions and therefore will never equal zero.======our only critical number is x = -1/2.this forms two intervals to test the sign for: (-ā, -1/2) and (-1/2, ā)test values into the 2nd derivative for (-ā, -1/2)if we try x = -1, we get
so the since the 2nd derivative of the curve is positive on the interval (-ā, -1/2), the curve is concave up on (-ā, -1/2)test values into the 2nd derivative for (-1/2, ā)if we try x = 0, we get
so the since the 2nd derivative of the curve is positive on the interval (-ā, -1/2), the curve is concave down on (-ā, -1/2).the interval on which the curve is concave up is (-ā, -1/2)