Find the Critical Points f(x)=3x(16-x)^3

Problem

ƒ(x)=3*x*(16−x)3

Solution

1. Apply the product rule to find the derivative of the function, where u=3*x and v=(16−x)3

ƒ(x)′=(d(3)*x)/d(x)*(16−x)3+3*xd(16−x)/d(x)

2. Differentiate each part, using the chain rule for the second term.

ƒ(x)′=3*(16−x)3+3*x*(3*(16−x)2*(−1))

3. Simplify the expression by multiplying the constants in the second term.

ƒ(x)′=3*(16−x)3−9*x*(16−x)2

4. Factor out the greatest common factor, which is 3*(16−x)2

ƒ(x)′=3*(16−x)2*((16−x)−3*x)

5. Combine like terms inside the parentheses to find the simplified derivative.

ƒ(x)′=3*(16−x)2*(16−4*x)

6. Set the derivative to zero to find the critical points.

3*(16−x)2*(16−4*x)=0

7. Solve for x by setting each factor equal to zero.

(16−x)2=0⇒x=16

16−4*x=0⇒4*x=16⇒x=4

Final Answer

x=4,16

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