Find Where Increasing/Decreasing Using Derivatives f(x)=x^4-32x+4

Problem

ƒ(x)=x4−32*x+4

Solution

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

d(ƒ(x))/d(x)=4*x3−32

2. Set the derivative to zero to find the critical points where the function might change direction.

4*x3−32=0

3. Solve for x by isolating the variable.

4*x3=32

x3=8

x=2

4. Test the intervals around the critical point x=2 to determine the sign of ƒ(x)′
   For x<2 test x=0

ƒ(0)′=4*(0)3−32=−32

Since ƒ(x)′<0 the function is decreasing on (−∞,2)

5. Test the interval for x>2
   For x>2 test x=3

ƒ(3)′=4*(3)3−32=108−32=76

Since ƒ(x)′>0 the function is increasing on (2,∞)

Final Answer

Increasing: *(2,∞), Decreasing: *(−∞,2)

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