question 12 (1 point)\nlet ( g(x) ) be a function with the following properties:\n( g(2)=5, g^{prime}(2)=12…

question 12 (1 point)\nlet ( g(x) ) be a function with the following properties:\n( g(2)=5, g^{prime}(2)=12, g^{prime prime}(2)=8 ).\nif ( f(x)=x^{3}-g(x) ), then which of the following is always true?\n( f(2) ) is an absolute minimum\n( f(2) ) is a local maximum\n( f(2) ) is a singular point\n( f(2) ) is a local minimum\n( f(2) ) is an absolute maximum

question 12 (1 point)\nlet ( g(x) ) be a function with the following properties:\n( g(2)=5, g^{prime}(2)=12, g^{prime prime}(2)=8 ).\nif ( f(x)=x^{3}-g(x) ), then which of the following is always true?\n( f(2) ) is an absolute minimum\n( f(2) ) is a local maximum\n( f(2) ) is a singular point\n( f(2) ) is a local minimum\n( f(2) ) is an absolute maximum

Answer

Explanation:

Step1: Find the first - derivative of (f(x))

Using the power rule ((x^n)^\prime=nx^{n - 1}) and the difference rule ((u - v)^\prime=u^\prime - v^\prime). If (f(x)=x^{3}-g(x)), then (f^\prime(x)=3x^{2}-g^\prime(x)). Evaluate (f^\prime(x)) at (x = 2): (f^\prime(2)=3\times(2)^{2}-g^\prime(2)). Substitute (g^\prime(2)=12) into the equation: (f^\prime(2)=3\times4 - 12=12 - 12=0).

Step2: Find the second - derivative of (f(x))

Using the power rule ((x^n)^\prime=nx^{n - 1}) and the difference rule ((u - v)^\prime=u^\prime - v^\prime) again. If (f^\prime(x)=3x^{2}-g^\prime(x)), then (f^{\prime\prime}(x)=6x - g^{\prime\prime}(x)). Evaluate (f^{\prime\prime}(x)) at (x = 2): (f^{\prime\prime}(2)=6\times2 - g^{\prime\prime}(2)). Substitute (g^{\prime\prime}(2)=8) into the equation: (f^{\prime\prime}(2)=12 - 8 = 4>0).

Step3: Use the second - derivative test

The second - derivative test states that if (f^\prime(c)=0) and (f^{\prime\prime}(c)>0), then (f(x)) has a local minimum at (x = c). Since (f^\prime(2)=0) and (f^{\prime\prime}(2)=4>0), (f(x)) has a local minimum at (x = 2).

Answer:

(f(2)) is a local minimum