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Chapter 4: Problem 37

For a function \(f, f^{\prime \prime}(x)>0\) for all real number input values. Describe the concavity of a graph of \(f\) and sketch a function for which this condition is true.

Short Answer

Expert verified
The graph of \( f \) is concave up everywhere.

Step by step solution

01

Understanding Concavity

The second derivative of a function, denoted by \( f''(x) \), explains the concavity of the function's graph. If \(f''(x) > 0\) for all \(x\), this implies that the graph of the function \(f\) is concave up for all values of \(x\). This means that the slope of the tangent line is increasing.
02

Drawing a Sample Function

To sketch a function where \(f''(x) > 0\), consider simple example functions, such as quadratic functions. The graph of \(f(x) = x^2\) is an upward-opening parabola where \(f''(x) = 2 > 0\) for all \(x\). Thus, it meets the condition that the second derivative is always positive, confirming the graph is concave up everywhere.

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Key Concepts

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

Second Derivative
The second derivative of a function, represented as \( f''(x) \), provides valuable insight into the shape of a function's graph. Understanding the second derivative can help you determine how the function's rate of change is changing.
  • If \( f''(x) > 0 \), the function is said to be concave up.
  • If \( f''(x) < 0 \), the function is concave down.
  • If \( f''(x) = 0 \), it might indicate a point of inflection, where the function could change concavity.
By analyzing \( f''(x) \), you gather information about whether the slope of a tangent to the function is increasing or decreasing. This can deeply inform the overall sketch and behavior of the function.
Concave Up
When we say a graph is concave up, it might help to picture it as a valley or a U-shape pattern. In mathematical terms, a graph is concave up if the second derivative is positive, \( f''(x) > 0 \).
The significance of a graph being concave up is:
  • The slope of the tangent line increases as you move along the x-axis.
  • The function value rises faster with respect to x.
  • The curve appears to open upward, much like a smile.
This upward-opening nature is why concave up graphs are often related to situations where there is acceleration in increase or growth.
Function Graph
The graph of a function visually represents how the output values (y-values) change in relation to input values (x-values). Identifying the concavity through its second derivative allows you to accurately sketch and interpret function graphs.
Important considerations for sketching or interpreting a function graph:
  • Use second derivative tests to determine whether sections of the graph are concave up or concave down.
  • Locate critical points and inflection points, where changes in concavity might occur, to gather a comprehensive picture of the function's behavior.
  • Consider end behavior 鈥� how the graph behaves as \( x \to \pm \infty \).
Effectively analyzing a function graph can give insights into trends and behaviors of real-world phenomena.
Quadratic Functions
Quadratic functions are a straightforward example often used to explain concavity through second derivatives.They are typically expressed in the form \( f(x) = ax^2 + bx + c \).
These functions create parabolic graphs that are either concave up or down:
  • If \( a > 0 \), the parabola opens upward, indicating a concave up shape.
  • If \( a < 0 \), the parabola opens downward, indicating a concave down shape.
For a basic quadratic function of \( f(x) = x^2 \), its second derivative \( f''(x) = 2 \) is positive, demonstrating constant concavity upwards.Understanding these characteristics allows you to quickly determine a quadratic function's properties.

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Most popular questions from this chapter

Sketch the graph of a function \(f\) such that all of the following statements are true. \- \(f\) has a relative minimum at \(x=3\). \- \(f\) has a relative maximum at \(x=-1\). \- \(f^{\prime}(x)>0\) for \(x<-1\) and \(x>3\) \- \(f^{\prime}(x)<0\) for \(-1

Boyle's Law for gases states that when the mass of a gas remains constant, the pressure \(p\) and the volume \(v\) of the gas are related by the equation \(p v=c,\) where \(c\) is a constant whose value depends on the gas. Assume that at a certain instant, the volume of a gas is 75 cubic inches and its pressure is 30 pounds per square inch. Because of compression of volume, the pressure of the gas is increasing by 2 pounds per square inch every minute. At what rate is the volume changing at this instant?

The body-mass index (BMI) of an individual who weighs \(w\) pounds and is \(h\) inches tall is given as $$ B=703 \frac{w}{h^{2}} $$ (Source: \(C D C)\) a. Write an equation showing the relationship between the body-mass index and weight of a woman who is 5 feet 8 inches tall. b. Construct a related-rates equation showing the interconnection between the rates of change with respect to time of the weight and the body-mass index. c. Consider a woman who weighs 160 pounds and is 5 feet 8 inches tall. If \(\frac{d w}{d t}=1\) pound per month, evaluate and interpret \(\frac{d B}{d t}\). d. Suppose a woman who is 5 feet 8 inches tall has a bodymass index of 24 points. If her body-mass index is decreasing by 0.1 point per month, at what rate is her weight changing?

For Activities 7 through \(18,\) write the first and second derivatives of the function. \(g(x)=e^{3 x}-\ln 3 x\)

For Activities 7 through \(18,\) write the first and second derivatives of the function. \(f(s)=32 s^{3}-2.1 s^{2}+7 s\)
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