find any critical numbers for f and then use the second derivative test to decide whether the critical…

find any critical numbers for f and then use the second derivative test to decide whether the critical number(s) lead to relative maxima or relative minima. if f(c)=0 or f(c) does not exist for a critical number c, then the second derivative test gives no information. in this case, use the first derivative test instead.\n\nf(x)=(x - 2)^4\n\nwhat is/are the critical number(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the critical number(s) is/are x = 2. (simplify your answer. use a comma to separate answers as needed.)\nb. there are no critical numbers.\n\nwhere are the relative extrema? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\n\na. there is a relative minimum at x = and a relative maximum at x = (simplify your answers. use a comma to separate answers as needed.)\nb. there is a relative maximum at x = . there are no relative minima. (simplify your answer. use a comma to separate answers as needed.)\n\nc. there is a relative minimum at x = 2. there are no relative maxima. (simplify your answer. use a comma to separate answers as needed.)\n\nd. there are no relative extrema.

find any critical numbers for f and then use the second derivative test to decide whether the critical number(s) lead to relative maxima or relative minima. if f(c)=0 or f(c) does not exist for a critical number c, then the second derivative test gives no information. in this case, use the first derivative test instead.\n\nf(x)=(x - 2)^4\n\nwhat is/are the critical number(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the critical number(s) is/are x = 2. (simplify your answer. use a comma to separate answers as needed.)\nb. there are no critical numbers.\n\nwhere are the relative extrema? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\n\na. there is a relative minimum at x = and a relative maximum at x = (simplify your answers. use a comma to separate answers as needed.)\nb. there is a relative maximum at x = . there are no relative minima. (simplify your answer. use a comma to separate answers as needed.)\n\nc. there is a relative minimum at x = 2. there are no relative maxima. (simplify your answer. use a comma to separate answers as needed.)\n\nd. there are no relative extrema.

Answer

Explanation:

Step1: Find the first derivative

Use the power rule ((u^n)^\prime = nu^{n - 1}u^\prime). Let (u=x - 2), (n = 4). Then (f^\prime(x)=4(x - 2)^3\times1=4(x - 2)^3). Set (f^\prime(x)=0), so (4(x - 2)^3=0). Solving for (x), we get (x = 2). So the critical number is (x = 2).

Step2: Find the second derivative

Differentiate (f^\prime(x)=4(x - 2)^3) using the power rule again. (f^{\prime\prime}(x)=12(x - 2)^2). Evaluate (f^{\prime\prime}(2)): (f^{\prime\prime}(2)=12(2 - 2)^2=0). Since the second - derivative test is inconclusive ((f^{\prime\prime}(c)=0)), we use the first - derivative test.

Step3: First - derivative test

Choose a test point to the left of (x = 2), say (x = 1). (f^\prime(1)=4(1 - 2)^3=-4<0). Choose a test point to the right of (x = 2), say (x = 3). (f^\prime(3)=4(3 - 2)^3=4>0). Since (f^\prime(x)) changes sign from negative to positive at (x = 2), there is a relative minimum at (x = 2).

Answer:

For the critical - number question: A. The critical number(s) is/are (x = 2). For the relative - extrema question: C. There is a relative minimum at (x = 2). There are no relative maxima.