section 03.5: problem 8\n(4 points)\ndetermine the interval(s) on which ( f(x)=4(x - 5)^{2}+5 ) is…

section 03.5: problem 8\n(4 points)\ndetermine the interval(s) on which ( f(x)=4(x - 5)^{2}+5 ) is increasing and decreasing.\n- increasing:\n- decreasing:

section 03.5: problem 8\n(4 points)\ndetermine the interval(s) on which ( f(x)=4(x - 5)^{2}+5 ) is increasing and decreasing.\n- increasing:\n- decreasing:

Answer

Explanation:

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

Using the chain rule, if (y = 4(u)^{2}+5) where (u=x - 5), then (\frac{dy}{du}=8u) and (\frac{du}{dx}=1). By the chain - rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). So (f^{\prime}(x)=8(x - 5)).

Step2: Find the critical point

Set (f^{\prime}(x)=0), then (8(x - 5)=0), which gives (x = 5).

Step3: Test intervals

  • For (x<5) (e.g., (x = 4)), (f^{\prime}(4)=8(4 - 5)=-8<0).
  • For (x>5) (e.g., (x = 6)), (f^{\prime}(6)=8(6 - 5)=8>0).

Answer:

  • Increasing: ((5,\infty))
  • Decreasing: ((-\infty,5))