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Chapter 5: Problem 7

\(f(x)=x^{4}-3 x^{3}+3 x^{2}+1\)

Short Answer

Expert verified
The function has a local minimum at \(x = 0\) with value \(f(0) = 1\).

Step by step solution

01

Understand the Function

Given the function is: \(f(x) = x^4 - 3x^3 + 3x^2 + 1\). This is a polynomial of degree 4.
02

Finding the Derivative

To find the critical points, first compute the first derivative of the function. The first derivative of \(f(x)\) is found using the power rule: \[ f'(x) = 4x^3 - 9x^2 + 6x \].
03

Setting the Derivative to Zero

Set the first derivative equal to zero to find the critical points: \[ 4x^3 - 9x^2 + 6x = 0 \].
04

Solving for Critical Points

Factor the equation \[ 4x^3 - 9x^2 + 6x = 0 \] to find: \[ x(4x^2 - 9x + 6) = 0 \]. Set each factor equal to zero: \[ x = 0 \] and solve \[ 4x^2 - 9x + 6 = 0 \]. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve \( 4x^2 - 9x + 6 = 0 \).
05

Quadratic Formula Application

For the quadratic equation \(4x^2 - 9x + 6 = 0\): \(a = 4\), \(b = -9\), and \(c = 6\). So: \[ x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 4 \cdot 6}}{2 \cdot 4} = \frac{9 \pm \sqrt{81 - 96}}{8} = \frac{9 \pm \sqrt{-15}}{8} \]. Since the discriminant is negative, there are no real solutions, and the only critical point is \(x = 0\).
06

Evaluate the Function at Critical Points

Evaluate the function \(f(x)\) at \(x = 0\): \[ f(0) = (0)^4 - 3(0)^3 + 3(0)^2 + 1 = 1 \]. Hence, the critical point is \(x = 0\).
07

Second Derivative Test

To determine the concavity at the critical point, find the second derivative of \(f(x)\): \[ f''(x) = 12x^2 - 18x + 6 \]. Evaluate the second derivative at \(x = 0\): \[ f''(0) = 12(0)^2 - 18(0) + 6 = 6 \]. Since \(f''(0) > 0\), the function has a local minimum at \(x = 0\).
08

Conclusion

The function \(f(x) = x^4 - 3x^3 + 3x^2 + 1\) has a local minimum at \(x = 0\). The minimum value is \(f(0) = 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
A polynomial function is a type of mathematical function that involves a sum of powers of its variable. In our exercise, the given polynomial function is:

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Most popular questions from this chapter

The function \(f\) is differentiable at each number in the closed interval \([a, b] .\) Prove that if \(f^{\prime}(a) \cdot f^{\prime}(b)<0\), there is a number \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\).

\(f(x)= \begin{cases}2(x-1)^{3} & \text { if } x<1 \\ (x-1)^{4} & \text { if } x \geq 1\end{cases}\)

Prove by the method of this section that the shortest distance from the point \(P_{1}\left(x_{1}, y_{1}\right)\) to the line \(l\), having the equation \(A x+B y+C=0\), is \(\left|A x_{1}+B y_{1}+C\right| / \sqrt{A^{2}+\bar{B}^{2}} .\) (HINT: If \(s\) is the number of units from \(P_{1}\) to a point \(P(x, y)\) on \(l\), then \(s\) will be an absolute minimum when \(s^{2}\) is an absolute minimum.)

\(G(x)=(x-5)^{2 / 3} ;(-\infty,+\infty)\)

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