find f(x) and find the value(s) of x where the tangent line is horizontal. f(x)=x^5(x - 12)^7 f(x)=□

Explanation:

Step1: Apply product - rule

The product - rule states that if (y = u\cdot v), then (y^\prime=u^\prime v + uv^\prime). Let (u = x^{5}) and (v=(x - 12)^{7}). First, find (u^\prime) and (v^\prime). The derivative of (u=x^{5}) using the power - rule ((x^{n})^\prime=nx^{n - 1}) is (u^\prime = 5x^{4}). The derivative of (v=(x - 12)^{7}) using the chain - rule ((f(g(x)))^\prime=f^\prime(g(x))\cdot g^\prime(x)) where (f(u)=u^{7}), (u = x - 12), (f^\prime(u)=7u^{6}) and (g^\prime(x)=1), so (v^\prime=7(x - 12)^{6}).

Step2: Calculate (f^\prime(x))

By the product - rule (f^\prime(x)=u^\prime v+uv^\prime), we have (f^\prime(x)=5x^{4}(x - 12)^{7}+x^{5}\cdot7(x - 12)^{6}). Factor out the common factors (x^{4}(x - 12)^{6}): [ \begin{align*} f^\prime(x)&=x^{4}(x - 12)^{6}[5(x - 12)+7x]\ &=x^{4}(x - 12)^{6}(5x-60 + 7x)\ &=x^{4}(x - 12)^{6}(12x-60)\ &=12x^{4}(x - 12)^{6}(x - 5) \end{align*} ]

Step3: Find when the tangent line is horizontal

The tangent line is horizontal when (f^\prime(x)=0). Set (f^\prime(x)=12x^{4}(x - 12)^{6}(x - 5)=0). Using the zero - product property (ab = 0) implies (a = 0) or (b = 0), we have:

• (x^{4}=0) gives (x = 0) (with multiplicity 4).
• ((x - 12)^{6}=0) gives (x = 12) (with multiplicity 6).
• (x - 5=0) gives (x = 5).

Answer:

(f^\prime(x)=12x^{4}(x - 12)^{6}(x - 5)); The values of (x) where the tangent line is horizontal are (x = 0), (x = 5), and (x = 12).