Answer each question appropriately.
-Find a curve $y = f ( x )$ with the following properties:
i. $\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } } = 8 \mathrm { x }$
ii. The graph passes through the point $( 0,1 )$ and has a horizontal tangent at that point.
A) $y = 8 x + 1$
B) $y = \frac { 4 } { 3 } x ^ { 3 } + x + 1$
C) $y = 4 x ^ { 2 } + 1$
D) $y = \frac { 4 } { 3 } x ^ { 3 } + 1$

Question

Quizplus · Accepted Answer

From the given equation, we can see that the first derivative of y is 4x^2 + C where C is a constant of integration. Taking the derivative again gives us the given second derivative of 8x.
Using the fact that the graph passes through (0,1) and has a horizontal tangent at that point, we can see that the constant C must be 0. Plugging this back into the first derivative gives us 4x^2, and integrating this gives us the function y = (4/3)x^3 + K, where K is a constant of integration.
Since the graph passes through (0,1), we know that K = 1, giving us the curve y = (4/3)x^3 + 1.

Answer Each Question Appropriately y=f(x)y = f ( x )y=f(x) With the Following Properties: I

Answer Each Question Appropriately $y = f ( x )$ With the Following Properties:
I