Find the Roots (Zeros) f(x)=5x^6+x^2 log of x+x

Problem

ƒ(x)=5*x6+x2*ln(x)+x

Solution

1. Identify the domain of the function. Because the expression contains ln(x) the variable x must be strictly greater than zero (x>0.

2. Set the function equal to zero to find the roots.

5*x6+x2*ln(x)+x=0

3. Factor out the common term x from the expression.

x*(5*x5+x*ln(x)+1)=0

4. Analyze the factors. Since x>0 due to the natural log domain, the factor x can never be zero. Therefore, any root must satisfy the equation:

5*x5+x*ln(x)+1=0

5. Evaluate the behavior of the remaining expression for x>0 For all x≥1 the terms 5*x5 x*ln(x) and 1 are all positive, so the sum is strictly greater than zero.

6. Check the interval 0<x<1 In this range, ln(x) is negative. However, as x approaches 0 from the right, 5*x5→0 x*ln(x)→0 and the constant term 1 remains. Thus, the function approaches 1

7. Determine if a sign change occurs. By calculating the derivative or testing values, it can be shown that 5*x5+x*ln(x)+1 remains positive for all x in the domain (0,∞)

8. Conclude that there are no values of x that satisfy the equation.

Final Answer

5*x6+x2*ln(x)+x=0⇒No real roots

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