let g(x) be twice differentiable function. which of the following statements about the function\n\nf(x)=g(x)+…

let g(x) be twice differentiable function. which of the following statements about the function\n\nf(x)=g(x)+x^{2}\n\nis true?\n\nif g(x)<-2 on 0,1, then f(x) is increasing on 0,1.\n\nif g(x)>0 on 1,10, then f(10) is the absolute maximum on 1,10.\n\nif g(0)=0 and g(0)>0, then x=0 is a local maximum of f(x).\n\nif g(x)<0 on -5,-1, then f(x) is increasing on -5,-1.\n\nif g(x)<-3 on (0,2), then f(0) is the absolute minimum on 0,2.

let g(x) be twice differentiable function. which of the following statements about the function\n\nf(x)=g(x)+x^{2}\n\nis true?\n\nif g(x)<-2 on 0,1, then f(x) is increasing on 0,1.\n\nif g(x)>0 on 1,10, then f(10) is the absolute maximum on 1,10.\n\nif g(0)=0 and g(0)>0, then x=0 is a local maximum of f(x).\n\nif g(x)<0 on -5,-1, then f(x) is increasing on -5,-1.\n\nif g(x)<-3 on (0,2), then f(0) is the absolute minimum on 0,2.

Answer

Explanation:

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

Given (f(x)=g(x)+x^{2}), by the sum - rule of differentiation (f^{\prime}(x)=g^{\prime}(x)+2x).

Step2: Analyze each option

  • Option 1: If (g^{\prime}(x)< - 2) on ([0,1]), then (f^{\prime}(x)=g^{\prime}(x)+2x). For (x\in[0,1]), (2x\in[0,2]). So (f^{\prime}(x)=g^{\prime}(x)+2x< - 2 + 2=0) on ([0,1]). So (f(x)) is decreasing on ([0,1]), this option is false.
  • Option 2: If (g^{\prime}(x)>0) on ([1,10]), then (f^{\prime}(x)=g^{\prime}(x)+2x>0) (since (x\in[1,10]) and (2x\geq2)). So (f(x)) is increasing on ([1,10]). Then (f(10)) is the absolute maximum on ([1,10]) (because if a function (y = f(x)) is increasing on ([a,b]), then (f(b)=\max_{x\in[a,b]}f(x))).
  • Option 3: If (g^{\prime}(0) = 0) and (g^{\prime\prime}(0)>0), then (f^{\prime}(x)=g^{\prime}(x)+2x), (f^{\prime}(0)=g^{\prime}(0)+0 = 0), and (f^{\prime\prime}(x)=g^{\prime\prime}(x)+2), (f^{\prime\prime}(0)=g^{\prime\prime}(0)+2>0). By the second - derivative test, (x = 0) is a local minimum of (f(x)) (since (f^{\prime}(0) = 0) and (f^{\prime\prime}(0)>0)), this option is false.
  • Option 4: If (g^{\prime}(x)<0) on ([-5,-1]), then (f^{\prime}(x)=g^{\prime}(x)+2x). For (x\in[-5,-1]), (2x\in[-10,-2]). So (f^{\prime}(x)=g^{\prime}(x)+2x<0) (sum of two negative numbers), (f(x)) is decreasing on ([-5,-1]), this option is false.
  • Option 5: If (g^{\prime}(x)<-3) on ((0,2)), then (f^{\prime}(x)=g^{\prime}(x)+2x). For (x\in(0,2)), (2x\in(0,4)). (f^{\prime}(x)) can be positive or negative. For example, when (x = 2), (f^{\prime}(2)=g^{\prime}(2)+4). If (g^{\prime}(2)=-3.5), then (f^{\prime}(2)=-3.5 + 4=0.5>0). So we cannot say (f(0)) is the absolute minimum on ([0,2]), this option is false.

Answer:

If (g^{\prime}(x)>0) on ([1,10]), then (f(10)) is the absolute maximum on ([1,10]).