Find the Critical Points y=x^3-4x^2-16x+5

Problem

y=x3−4*x2−16*x+5

Solution

1. Find the derivative of the function with respect to x to determine the slope of the tangent line.

d(y)/d(x)=3*x2−8*x−16

2. Set the derivative to zero to find the xvalues where the slope is horizontal, which defines the critical points.

3*x2−8*x−16=0

3. Factor the quadratic equation to solve for x

(3*x+4)*(x−4)=0

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

3*x+4=0⇒x=−4/3

x−4=0⇒x=4

5. Find the y-coordinates by substituting the xvalues back into the original function.

y*(−4/3)=(−4/3)3−4*(−4/3)2−16*(−4/3)+5=499/27

y(4)=(4)3−4*(4)2−16*(4)+5=−59

Final Answer

Critical Points:(−4/3,499/27),(4,−59)

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