Problem 103
Question
Show that the function \(f(x)=x^{3}+x+1\) has no relative extrema on \((-\infty, \infty)\).
Step-by-Step Solution
Verified Answer
The function \(f(x) = x^3 + x + 1\) has no relative extrema on \((-\infty, \infty)\) because its first derivative, \(f'(x) = 3x^2 + 1\), is always positive and defined for all real values of x, yielding no critical points.
1Step 1: Find the first derivative of the function
To find the critical points of the function, we need to find its first derivative. The function is \(f(x) = x^3 + x + 1\). Apply the power rule:
\(f'(x) = 3x^2 + 1\)
2Step 2: Determine if the derivative has any points where it is equal to zero or undefined
We now need to analyze the first derivative that we found in the previous step to find any critical points.
\(f'(x) = 3x^2 + 1\)
Since \(f'(x)\) is a quadratic function that does not contain any fraction or radical, it is defined for all values of x. Therefore, there are no points where \(f'(x)\) is undefined.
Next, we need to determine if \(f'(x)\) has any points where it is equal to zero. To do this, set \(f'(x)\) equal to zero and solve for x:
\(3x^2 + 1 = 0\)
However, in this case, there are no real solutions for x because for any real value of x, \(3x^2\) will always be non-negative, so \(3x^2 + 1\) will always be greater than 0. Since we cannot find any real values of x that make \(f'(x) = 0\), there are no critical points for this function.
3Step 3: Conclusion
Since we could not find any critical points of the function \(f(x) = x^3 + x + 1\), we can conclude that it has no relative extrema on \((-\infty, \infty)\).
Key Concepts
First Derivative TestCritical PointsPower Rule
First Derivative Test
The First Derivative Test is a crucial mathematical process used to determine the relative extrema of a function. When studying the behavior of a function, one of the questions often asked is where the function reaches its highest or lowest points, known as relative maxima or minima. To apply the First Derivative Test, we must first locate the function's critical points, which are points where the function's derivative is either zero or undefined.
Once the critical points are found, these points are analyzed to understand how the function's slope changes. If the derivative changes from positive to negative at a critical point, the point is a relative maximum. If the derivative changes from negative to positive, the point is a relative minimum. If there is no change, the test is inconclusive.
For the given function, \( f(x) = x^3 + x + 1 \), the derivative \( f'(x) = 3x^2 + 1 \) never equals zero, indicating there are no points at which the slope of the original function changes from positive to negative or vice versa. As such, no relative maxima or minima exist for this function, signaling a consistently increasing function.
Once the critical points are found, these points are analyzed to understand how the function's slope changes. If the derivative changes from positive to negative at a critical point, the point is a relative maximum. If the derivative changes from negative to positive, the point is a relative minimum. If there is no change, the test is inconclusive.
For the given function, \( f(x) = x^3 + x + 1 \), the derivative \( f'(x) = 3x^2 + 1 \) never equals zero, indicating there are no points at which the slope of the original function changes from positive to negative or vice versa. As such, no relative maxima or minima exist for this function, signaling a consistently increasing function.
Critical Points
Critical points are where a function's derivative is either zero or does not exist. These points are essential in analyzing the function's graph because they can potentially indicate the location of relative extrema like peaks or troughs. Finding the critical points is often one of the first steps in studying a function's behavior.
To find the critical points, one must calculate the derivative of the function and solve for the values of the variable that make the derivative zero or identify where the derivative is undefined. In the context of our function \( f(x) = x^3 + x + 1 \), the derivative \( f'(x) = 3x^2 + 1 \) does not have any real roots, meaning it is never zero, and it is also never undefined since it is a simple polynomial. Therefore, no critical points can be identified, and thus no relative extrema exist for this function over the entire set of real numbers.
To find the critical points, one must calculate the derivative of the function and solve for the values of the variable that make the derivative zero or identify where the derivative is undefined. In the context of our function \( f(x) = x^3 + x + 1 \), the derivative \( f'(x) = 3x^2 + 1 \) does not have any real roots, meaning it is never zero, and it is also never undefined since it is a simple polynomial. Therefore, no critical points can be identified, and thus no relative extrema exist for this function over the entire set of real numbers.
Power Rule
The Power Rule is one of the most fundamental rules of differentiation, which is used to find the derivative of a function with respect to a variable. This rule applies to monomials where a real number is raised to a power. The Power Rule states that if the function is \( f(x) = ax^n \), then the derivative of the function, \( f'(x) \), is \( anx^{n-1} \).
Applying the Power Rule simplifies finding derivatives significantly, particularly for polynomial functions. For instance, in the function \( f(x) = x^3 + x + 1 \), applying the Power Rule to each term individually, we get the derivative \( f'(x) = 3x^2 + 1 \) (since the derivative of a constant is zero and the derivative of \( x \) is 1). This rule proves to be invaluable when analyzing the function for critical points as part of determining the locations of relative extrema.
Applying the Power Rule simplifies finding derivatives significantly, particularly for polynomial functions. For instance, in the function \( f(x) = x^3 + x + 1 \), applying the Power Rule to each term individually, we get the derivative \( f'(x) = 3x^2 + 1 \) (since the derivative of a constant is zero and the derivative of \( x \) is 1). This rule proves to be invaluable when analyzing the function for critical points as part of determining the locations of relative extrema.
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