Determine all values of x, (if any), at which the graph of the function has a horizontal tangent.
B) and
C) and
E)The graph has no horizontal tangents.

Question

Determine all values of x, (if any), at which the graph of the function has a horizontal tangent. 
B) and 
C) and 
E)The graph has no horizontal tangents.

Quizplus · Accepted Answer

To find where the graph has a horizontal tangent, we need to find where the derivative is equal to zero. Taking the derivative of the function, we get f'(x) = 4x^3 - 12x^2 - 16x + 12. Setting this equal to zero and solving for x, we get x = 1, 2, or 3/2. These are the possible x-values where the graph could have a horizontal tangent.

To confirm if these points actually have horizontal tangents, we can check the sign of the derivative on either side of each point. Plugging in a value slightly less than 1, we get f'(-1) = -32, which is negative. Plugging in a value slightly greater than 1, we get f'(1.1) = 5.576, which is positive. Therefore, at x = 1, the graph has a horizontal tangent.

Similarly, we can check the other two possible points. At x = 2, the graph also has a horizontal tangent, but at x = 3/2, the derivative changes sign and therefore there is no horizontal tangent at that point.

Thus, the answer is B) 1 and 2.

Determine All Values of X, (If Any), at Which the Graph