Question

Please help me with this question. I’m stuck

Answer 1

Answer:

(6,1)

Step-by-step explanation:

To find the points of inflection, we need to find where the second derivative is equal to zero, or does not exist.

I will take the derivative using the chain rule.

$f(x) = 5\sqrt[3]{(x-6) } + 1 \\\\f(x) = 5(x-6)^{\frac{1}{3} } + 1$

The first derivative:

$f'(x) = 5*\frac{1}{3} (x-6)^{-2/3} \\\\f'(x) = \frac{5}{3} (x-6)^{-2/3}$

The second derivative:

$f'(x) = \frac{5}{3} (x-6)^{-2/3} \\\\f''(x) = \frac{5}{3}*-\frac{2}{3} (x-6)^{-5/3}\\\\ f''(x) = -\frac{10}{9} (x-6)^{-5/3}$

Now to find the inflection points we have to find where the second derivative is equal to zero, or do not exist.

$f''(x) = -\frac{10}{9} (x-6)^{-5/3}\\0= -\frac{10}{9} (x-6)^{-5/3}\\0= (x-6)^{-5/3}$

We can see that the second derivative does not exist when x=6, so there is an inflection point there.

We can solve the original equation to find the coordinate for x = 6.

f(x) = 5∛(x-6) + 1

f(6) = 5∛(6-6) + 1

f(6) =  1

So there is an inflection point at (6,1)