the function ( y = f(x) ) is continuous on ( (-infty, infty) ). use the sign chart of ( f(x) ) to determine…

the function ( y = f(x) ) is continuous on ( (-infty, infty) ). use the sign chart of ( f(x) ) to determine the relative extrema of ( y = f(x) ).\n\n(description: first derivative number line sign chart. ( f(x) ) to the left and above the number line, ( x ) to the left and below the number line. three tick marks on the number line. below the tick marks are numbers -3, 2, and 9. corresponding above -3 is und, above 2 is 0, and above 9 is also zero. above number line, denoted with pluses from the left to und, with minuses between -3 and 2, minuses between 2 and 9; and pluses from 9 to the right.)\n\nrelative minimum at ( x = -3 ); and\n\nrelative maximum at ( x = 9 )\n\nrelative maximum at ( x = -3 ); and\n\nrelative minimum at ( x = 9 )\n\nrelative minimum at ( x = 2 ).\n\nrelative minimum at ( x = 2 ) and ( x = 9 ).\n\nno correct answer choice is given.

the function ( y = f(x) ) is continuous on ( (-infty, infty) ). use the sign chart of ( f(x) ) to determine the relative extrema of ( y = f(x) ).\n\n(description: first derivative number line sign chart. ( f(x) ) to the left and above the number line, ( x ) to the left and below the number line. three tick marks on the number line. below the tick marks are numbers -3, 2, and 9. corresponding above -3 is und, above 2 is 0, and above 9 is also zero. above number line, denoted with pluses from the left to und, with minuses between -3 and 2, minuses between 2 and 9; and pluses from 9 to the right.)\n\nrelative minimum at ( x = -3 ); and\n\nrelative maximum at ( x = 9 )\n\nrelative maximum at ( x = -3 ); and\n\nrelative minimum at ( x = 9 )\n\nrelative minimum at ( x = 2 ).\n\nrelative minimum at ( x = 2 ) and ( x = 9 ).\n\nno correct answer choice is given.

Answer

Explanation:

Step1: Recall the first - derivative test

If (f^{\prime}(x)) changes sign from positive to negative at a critical point (x = c), then (f(x)) has a relative maximum at (x = c). If (f^{\prime}(x)) changes sign from negative to positive at a critical point (x = c), then (f(x)) has a relative minimum at (x = c). A critical point is a point where (f^{\prime}(x)=0) or (f^{\prime}(x)) is undefined.

Step2: Analyze the sign changes

  • At (x=-3): (f^{\prime}(x)) changes from positive (to the left of (x = - 3)) to negative (to the right of (x=-3)). So (x=-3) is a relative maximum.
  • At (x = 2): (f^{\prime}(x)) does not change its sign (it is negative on both sides of (x = 2)). So (x = 2) is not a relative extremum.
  • At (x=9): (f^{\prime}(x)) changes from negative (to the left of (x = 9)) to positive (to the right of (x = 9)). So (x = 9) is a relative minimum.

Answer:

Relative maximum at (x=-3); and Relative minimum at (x = 9) (the second option)