Question

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 \(g^\prime(x)=1\) is \(v^\prime=7(x - 12)^{6}\).

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

By the product - rule \(f^\prime(x)=u^\prime v+uv^\prime\), so \(f^\prime(x)=5x^{4}(x - 12)^{7}+x^{5}\cdot7(x - 12)^{6}\).
Factor out the common factors \(x^{4}(x - 12)^{6}\):

$$
LATEXBLOCK0
$$

Step3: Find where 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 \(ab = 0\) if and only if \(a = 0\) or \(b = 0\), we have:
\(x^{4}=0\Rightarrow x = 0\) (with multiplicity 4), \((x - 12)^{6}=0\Rightarrow x = 12\) (with multiplicity 6), and \(x - 5=0\Rightarrow x = 5\).

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 \(x = 12\).