Provide an appropriate response.
-Use the first derivative test to determine the local extrema, if any, for the function:
A) f(x) has a local maximum at x = 4.
B) f(x) has a local minimum at x = 4.
C) f(x) has no local extrema
D) f(x) has a local minimum at 6

Question

Provide an appropriate response.
-Use the first derivative test to determine the local extrema, if any, for the function: 
A) f(x) has a local maximum at x = 4. 
B) f(x) has a local minimum at x = 4. 
C) f(x) has no local extrema 
D) f(x) has a local minimum at 6

Quizplus · Accepted Answer

By using the first derivative test, we can find the critical points of the function. Taking the derivative of the function, we get 3x^2 - 24x + 35. Setting this equal to zero, we find the critical points to be x=5/3 and x=7/3. Using the first derivative test, we can determine that at x=4, the function has a local minimum since the function is decreasing to the left of x=4 and increasing to the right of x=4. Therefore, the correct answer is B, f(x) has a local minimum at x=4. The choice D is incorrect because there is no local extremum at x=6. Using the first derivative test, we find that the critical point of the function is at x = 4. To determine the nature of the critical point, we look at the sign of the derivative on either side of x = 4. The derivative is positive to the left of x = 4 and negative to the right of x = 4, so the function has a local minimum at x = 4. Therefore, the best choice is B, f(x) has a local minimum at x = 4.

Provide an Appropriate Response