Question

If the interval (a, infinity) describes all values of x for which the graph of f(x)=4/x^2-6x+9 is decreasing, what is the value of a

Answer 1

3

Explanation:

Answer 2

The answer is "3".

Explanation:

`\texttt{Given graph equation: } \\\\\bold{\Rightarrow f(x)=(4)/(x^2-6x+9) }`

`\bold{\ interval: (a,\infty)}`

differentiate the function f(x):

`\Rightarrow f'(x) = (d )/(dx)((4)/(x^2-6x+9))\\\\`

Formula:

`\bold{(d)/(dx) (v)/(u)= (u (d)/(dx) v- v(d)/(dx)u )/(u^2)}`

`\Rightarrow f'(x) = (d )/(dx)((4)/(x^2-6x+9))\\\\\\\Rightarrow f'(x) = (d )/(dx)(((x^2-6x+9) (d)/(dx) 4- 4(d)/(dx)(x^2-6x+9) )/((x^2-6x+9)^2))\\\\\\\Rightarrow f'(x) = (d )/(dx)(((x^2-6x+9) * 0 - 4(2x-6) )/((x^2-6x+9)^2))\\\\\\\Rightarrow f'(x) = (- 4(2x-6))/((x^2-6x+9)^2)\\\\\\\Rightarrow (- 4(2x-6))/((x^2-6x+9)^2)=0\\\\\Rightarrow - 4(2x-6)=0\\\\\Rightarrow 8x-24=0\\\\\Rightarrow 8x=24\\\\\Rightarrow x=(24)/(8)\\\\\Rightarrow x=3\\\\`

since the value of x in:
`(-3 ,\infty) \ \ and \ \ (3, \infty)` and
`f'(x) <= 0\\`. So, the value of a is 3