Graph f(x)=x^4-4x^3+10

Problem

ƒ(x)=x4−4*x3+10

Solution

1. Find the first derivative to identify critical points and determine where the function is increasing or decreasing.

d(ƒ(x))/d(x)=4*x3−12*x2

2. Set the first derivative to zero to find the critical values of x

4*x2*(x−3)=0

x=0,x=3

3. Determine the intervals of increase and decrease by testing values around the critical points.

ƒ(x)′<0* for *x∈(−∞,0)∪(0,3)

ƒ(x)′>0* for *x∈(3,∞)

4. Find the second derivative to determine the concavity and identify inflection points.

d2(ƒ(x))/(d(x)2)=12*x2−24*x

5. Set the second derivative to zero to find potential inflection points.

12*x*(x−2)=0

x=0,x=2

6. Analyze concavity using the second derivative.

ƒ(x)″>0* for *x∈(−∞,0)∪(2,∞)

ƒ(x)″<0* for *x∈(0,2)

7. Calculate key coordinates for plotting the graph.

ƒ(0)=10

ƒ(2)=−6

ƒ(3)=−17

8. Identify the end behavior of the polynomial.

(lim_x→∞)(ƒ(x))=∞

(lim_x→−∞)(ƒ(x))=∞

Final Answer

The graph of ƒ(x)=x4−4*x3+10 has a horizontal tangent and inflection point at (0,10) an inflection point at (2,−6) and a local minimum at (3,−17)

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