Example 14 Find and classify the critical points of $$ f(x)=x^{4}-4 x^{3}+5 $$.

Question

UpStudy AI · Accepted Answer

**Step 1. Find the Critical Points**

We begin with the function
$$
f(x)=x^4-4x^3+5.
$$
Critical points occur where the first derivative is zero or undefined. Since $$ f(x) $$ is a polynomial, $$ f'(x) $$ exists everywhere.

Differentiate $$ f(x) $$:
$$
f'(x)=\frac{d}{dx}(x^4)-\frac{d}{dx}(4x^3)+\frac{d}{dx}(5)=4x^3-12x^2.
$$

Factor the derivative:
$$
f'(x)=4x^2(x-3).
$$

Set the derivative equal to zero:
$$
4x^2(x-3)=0.
$$

This yields:
$$
4x^2=0 \quad \text{or} \quad x-3=0,
$$
which gives:
$$
x=0 \quad \text{or} \quad x=3.
$$

Thus, the critical points are at $$ x=0 $$ and $$ x=3 $$.

---

**Step 2. Classify the Critical Points**

We use the second derivative to classify the critical points.

Differentiate $$ f'(x)=4x^3-12x^2 $$ to obtain $$ f''(x) $$:
$$
f''(x)=\frac{d}{dx}(4x^3)-\frac{d}{dx}(12x^2)=12x^2-24x.
$$

Factor $$ f''(x) $$:
$$
f''(x)=12x(x-2).
$$

*For $$ x=3 $$:*

Evaluate:
$$
f''(3)=12\cdot3\,(3-2)=12\cdot3\cdot1=36.
$$
Since $$ f''(3)>0 $$, by the second derivative test, $$ x=3 $$ is a local minimum.

*For $$ x=0 $$:*

Evaluate:
$$
f''(0)=12\cdot0\,(0-2)=0.
$$
Since the second derivative is zero at $$ x=0 $$, the test is inconclusive. We must investigate further.

**Further Analysis at $$ x=0 $$:**

Look at the sign of $$ f'(x)=4x^2(x-3) $$ around $$ x=0 $$. Choose a small negative value, say $$ x=-0.1 $$, and a small positive value, say $$ x=0.1 $$:

- For $$ x=-0.1 $$:
  $$
  f'(-0.1)=4\cdot(-0.1)^2\,(-0.1-3)=4\cdot0.01\cdot(-3.1) \approx -0.124.
  $$
  
- For $$ x=0.1 $$:
  $$
  f'(0.1)=4\cdot(0.1)^2\,(0.1-3)=4\cdot0.01\cdot(-2.9) \approx -0.116.
  $$

In both cases, $$ f'(x) $$ is negative. Thus, there is no change in sign of the derivative as we pass through $$ x=0 $$. Consequently, the critical point at $$ x=0 $$ is neither a local maximum nor a local minimum. It is a stationary inflection point.

---

**Final Answer**

- $$ x=3 $$ is a local minimum.
- $$ x=0 $$ is a stationary inflection point (critical point that is not a local extremum).
The critical points are at $$ x=3 $$ (local minimum) and $$ x=0 $$ (stationary inflection point).

Example 14 Find and classify the critical points of \( f(x)=x^{4}-4 x^{3}+5 \).

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Calculus Questions