Find the Critical Points f(x)=3x^5-10x^3+15x

Problem

ƒ(x)=3*x5−10*x3+15*x

Solution

1. Find the derivative of the function ƒ(x) with respect to x using the power rule.

d(ƒ(x))/d(x)=15*x4−30*x2+15

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

15*x4−30*x2+15=0

3. Factor out the greatest common factor from the equation to simplify the expression.

15*(x4−2*x2+1)=0

4. Divide both sides by 15.

x4−2*x2+1=0

5. Factor the quadratic form expression, noting that it is a perfect square trinomial.

(x2−1)2=0

6. Solve for x by taking the square root of both sides and then factoring the resulting difference of squares.

x2−1=0

(x−1)*(x+1)=0

7. Identify the roots which represent the x-coordinates of the critical points.

x=1

x=−1

Final Answer

Critical Points: *x=1,x=−1

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