Final answer:
The function f(x)=-x^4+2x^3+15x^2 has a positive value and a positive slope at x = 3; however, more analysis is needed to determine if the slope's magnitude is decreasing with increasing x.
Step-by-step explanation:
The student is asking about the properties of a function represented by f(x)=-x^4+2x^3+15x^2. We can analyze this function to provide an accurate statement about its value and the behavior of its slope at a particular point, which is x = 3.
First, to evaluate the value of the function at x = 3, we substitute 3 into the function:
f(3) = -(3)^4 + 2(3)^3 + 15(3)^2
= -81 + 54 + 135
= 108
Since 108 is positive, we can confirm that at x = 3, the function has a positive value.
Next, we find the first derivative of the function to analyze the slope:
f'(x) = -4x^3 + 6x^2 + 30x
Then we evaluate it at x = 3:
f'(3) = -4(3)^3 + 6(3)^2 + 30(3)
= -108 + 54 + 90
= 36
Since 36 is positive, the function has a positive slope at x = 3. To check if the magnitude of slope is decreasing, we would look at the second derivative or simply inspect the change in slope around x = 3. However, without more information or analysis, we cannot definitively say whether it is decreasing in magnitude with increasing x solely based on the original function provided.