Question

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

• increasing:
• decreasing:

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: \((-\infty,5)\)