for a function f you are given that f(x)=4x - x². the function f is decreasing on the interval k, ∞. find…

Explanation:

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

Given (f^{\prime}(x)=4x - x^{2}), we want to find where (f^{\prime}(x)\leq0) (since (f(x)) is decreasing when (f^{\prime}(x)\leq0)). Set (4x - x^{2}\leq0), factor it as (x(4 - x)\leq0).

Step2: Solve the inequality (x(4 - x)\leq0)

Find the roots of (y = x(4 - x)=-x^{2}+4x). Using the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}) for (y = ax^{2}+bx + c) (here (a=-1), (b = 4), (c = 0)), we get (x=\frac{-4\pm\sqrt{16-0}}{-2}=\frac{-4\pm4}{-2}). The roots are (x = 0) and (x = 4). We can also test intervals:

• For (x<0), let (x=-1), then (y=(-1)(4+1)=-5<0).
• For (0<x<4), let (x = 1), then (y=(1)(4 - 1)=3>0).
• For (x>4), let (x = 5), then (y=(5)(4 - 5)=-5<0).

The inequality (x(4 - x)\leq0) is satisfied when (x\leq0) or (x\geq4).

Since we are looking at the interval ([k,\infty)), the smallest value of (k) for which (f(x)) is decreasing on ([k,\infty)) is (k = 4).

Answer:

(4)