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)=-x² + 2x + 24\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\n(simplify your answer. use a comma to separate answers as needed.)\n\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. the function has a relative minimum at x=\n\n. there are no relative maxima.\n\n(simplify your answer. use a comma to separate answers as needed.)\n\nb. the function has a relative maximum at x=\n\n. there are no relative minima.\n\n(simplify your answer. use a comma to separate answers as needed.)\n\nc. the function has a relative minimum at x=\n\nand a relative maximum at x=\n\n(simplify your answers. 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)=-x² + 2x + 24\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\n(simplify your answer. use a comma to separate answers as needed.)\n\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. the function has a relative minimum at x=\n\n. there are no relative maxima.\n\n(simplify your answer. use a comma to separate answers as needed.)\n\nb. the function has a relative maximum at x=\n\n. there are no relative minima.\n\n(simplify your answer. use a comma to separate answers as needed.)\n\nc. the function has a relative minimum at x=\n\nand a relative maximum at x=\n\n(simplify your answers. use a comma to separate answers as needed.)\n\nd. there are no relative extrema.

Answer

Explanation:

Step1: Find the first derivative

The function is ( f(x)=-x^{2}+2x + 24 ). Using the power rule ( (x^{n})^\prime=nx^{n - 1} ), we have ( f^\prime(x)=-2x+2 ). Set ( f^\prime(x) = 0 ), then ( -2x + 2=0 ). Solve for ( x ): [ \begin{align*} -2x&=-2\ x& = 1 \end{align*} ] So the critical number is ( x = 1 ).

Step2: Find the second derivative

Differentiate ( f^\prime(x)=-2x + 2 ) with respect to ( x ). Using the power rule, ( f^{\prime\prime}(x)=-2). Since ( f^{\prime\prime}(1)=-2<0 ), by the second - derivative test, when ( f^{\prime}(c) = 0 ) and ( f^{\prime\prime}(c)<0 ), the function ( y = f(x) ) has a relative maximum at ( x=c ).

Answer:

  • The critical number(s) is/are ( x = 1 ) (Option A).
  • The function has a relative maximum at ( x = 1 ) (Option B).