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 - 7)^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=\n(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=\n(simplify your answers. use a comma to separate answers as needed.)\nand a relative maximum at x=\n(simplify your answer. use a comma to separate answers as needed.)\nb. there is a relative minimum at x=\n(simplify your answer. use a comma to separate answers as needed.)\n. there are no relative maxima.\nc. there is a relative maximum at x=\n(simplify your answer. use a comma to separate answers as needed.)\n. there are no relative minima.\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 - 7)^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=\n(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=\n(simplify your answers. use a comma to separate answers as needed.)\nand a relative maximum at x=\n(simplify your answer. use a comma to separate answers as needed.)\nb. there is a relative minimum at x=\n(simplify your answer. use a comma to separate answers as needed.)\n. there are no relative maxima.\nc. there is a relative maximum at x=\n(simplify your answer. use a comma to separate answers as needed.)\n. there are no relative minima.\nd. there are no relative extrema.

Answer

Explanation:

Step1: Find the first derivative

Use the power rule ((x^n)^\prime = nx^{n - 1}). For (y=(x - 7)^4), let (u=x - 7), then (y = u^4). By the chain rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). (\frac{dy}{du}=4u^{3}) and (\frac{du}{dx}=1), so (f^\prime(x)=4(x - 7)^{3}). Set (f^\prime(x)=0), (4(x - 7)^{3}=0), which gives (x = 7). So the critical number is (x = 7).

Step2: Find the second derivative

Differentiate (f^\prime(x)=4(x - 7)^{3}) again. Using the chain rule, let (v=(x - 7)), (y = 4v^{3}). (\frac{dy}{dv}=12v^{2}) and (\frac{dv}{dx}=1), so (f^{\prime\prime}(x)=12(x - 7)^{2}). Evaluate (f^{\prime\prime}(x)) at (x = 7), (f^{\prime\prime}(7)=12(7 - 7)^{2}=0). Since the second - derivative test is inconclusive, use the first - derivative test. For (x<7) (e.g., (x = 6)), (f^\prime(6)=4(6 - 7)^{3}=-4<0). For (x>7) (e.g., (x = 8)), (f^\prime(8)=4(8 - 7)^{3}=4>0). The function changes from decreasing to increasing at (x = 7).

Answer:

  • Critical number: A. The critical number(s) is/are (x = 7)
  • Relative extrema: A. There is a relative minimum at (x = 7)