Find the Critical Points g(x)=x^7-7x^5

Problem

g(x)=x7−7*x5

Solution

1. Find the derivative of the function g(x) using the power rule.

d(g(x))/d(x)=7*x6−35*x4

2. Set the derivative equal to zero to find the critical points where the slope of the tangent line is horizontal.

7*x6−35*x4=0

3. Factor out the greatest common factor from the equation to isolate the variables.

7*x4*(x2−5)=0

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

7*x4=0⇒x=0

x2−5=0⇒x2=5⇒x=±√(,5)

Final Answer

x=0,√(,5),−√(,5)

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