Question

Find dy/dx and d2y/dx2. x = t2 + 5, y = t2 + 5t dy dx = Correct: Your answer is correct. d2y dx2 = Correct: Your answer is correct. For which values of t is the curve concave upward? (Enter your answer using interval notation.) Changed: Your submitted answer was incorrect. Your current answer has not been submitted.

Answer 1

To find the concavity of the curve, we need to find the second derivative, d2y/dx2. The derivative of y with respect to x is (2t + 5) / (2t), and the second derivative is -5 / (2t)^2. The curve is concave upward on the intervals (-∞,0) and (0,∞).

Step-by-step explanation:

To find the concavity of the curve, we need to find the second derivative, d2y/dx2. Given that x = t2 + 5 and y = t2 + 5t, we first need to find dy/dx by taking the derivative of y with respect to t and then dividing by the derivative of x with respect to t. From there, we can find the second derivative by differentiating dy/dx with respect to t and then dividing by the derivative of x with respect to t again.

First, let's find dy/dx:

dy/dx = (dy/dt) / (dx/dt)

Using the chain rule, we can find that dy/dx = (2t + 5) / (2t).

Next, let's find d2y/dx2:

d2y/dx2 = ((d2y/dt2) / (dx/dt)) - ((dy/dt)(d2x/dt2)) / (dx/dt)2

Using the chain rule, we can find that d2y/dx2 = -5 / (2t)2.

To find when the curve is concave upward, we need to find when d2y/dx2 > 0. Therefore, we need to find the values of t that satisfy the inequality:

-5 / (2t)2 > 0

Since t cannot equal 0 (as it would result in an undefined value), we can conclude that the curve is concave upward for all values of t except 0. Therefore, the curve is concave upward on the intervals (-∞,0) and (0,∞).