Problem 72 Determine the interval(s) on whi... [FREE SOLUTION]

91影视

Precalculus: An Investigation of Functions

David Lippman

$Math Studyset 91影视 Explanations$ Math

2017 Edition

Chapter 1: Problem 72

Determine the interval(s) on which the function is concave up and concave down. $$ b(x)=\sqrt[3]{-x-6} $$

Short Answer

Expert verified

The function is concave down on $(- \infty, -6)$ and concave up on $(-6, \infty)$.

Step by step solution

Understand the function and rewrite it for simplicity

The function given is $ b(x) = \sqrt[3]{-x-6} $. This can also be expressed using exponents as $ b(x) = (-x - 6)^{1/3} $.

Compute the first derivative $ b'(x) $

To find the first derivative, apply the chain rule to $ b(x) = (-x - 6)^{1/3} $. The derivative is: \[ b'(x) = \frac{1}{3}(-x - 6)^{-2/3}(-1) = -\frac{1}{3}(-x - 6)^{-2/3}. \]

Compute the second derivative $ b''(x) $

Differentiate the first derivative to obtain the second derivative: \[ b''(x) = \frac{d}{dx}\left(-\frac{1}{3}(-x - 6)^{-2/3}\right). \]Using the chain rule again, \[ b''(x) = \frac{2}{9}(-x - 6)^{-5/3}\cdot(-1) = -\frac{2}{9}(-x - 6)^{-5/3}. \]

Determine the critical points from the second derivative

Since $ b''(x) = -\frac{2}{9}(-x - 6)^{-5/3} $, it is undefined when $-x - 6 = 0$, i.e., at $ x = -6 $. This is where the concavity could change.

Test intervals around critical point to determine concavity

Test values to the left and right of $ x = -6 $:- For $ x < -6 $, choose $ x = -7 $: \[ b''(-7) = -\frac{2}{9}(-(-7) - 6)^{-5/3} = -\frac{2}{9}(1)^{-5/3} = -\frac{2}{9}. \] This is negative, indicating the function is concave down.- For $ x > -6 $, choose $ x = -5 $: \[ b''(-5) = -\frac{2}{9}(-(-5) - 6)^{-5/3} = -\frac{2}{9}(-1)^{-5/3} = \frac{2}{9}. \] This is positive, indicating the function is concave up.

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Key Concepts

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

first derivative

A first derivative is a key concept in calculus that represents the rate at which a function is changing at any given point. In simpler terms, it tells us the slope of the function's curve. For the given function, $ b(x) = (-x - 6)^{1/3} $, finding the first derivative involves applying the chain rule. This is because the function can be rewritten as a power of $-x - 6$. Here, we differentiate the power and then the expression inside the power separately.

Start with the outer function, the cube root, which becomes a fractional power (1/3).
Apply the chain rule: differentiate the inside $-x - 6$, which results in multiplying by $-1$.
Combine these to get $-\frac{1}{3}(-x-6)^{-2/3}$.

This outcome shows how the function's slope behaves, which is instrumental for discussing increasing/decreasing functions and predicting concavity.

second derivative

The second derivative of a function is derived from taking the derivative of the first derivative. It provides us with valuable information about the concavity of the function. For the function $ b(x) = (-x-6)^{1/3} $, once the first derivative $ b'(x) = -\frac{1}{3}(-x - 6)^{-2/3} $ is obtained, we proceed to find the second derivative using the same chain rule.

The second derivative tells us how the rate of change itself is changing.
To obtain $ b''(x) $, differentiate $ -\frac{1}{3}(-x - 6)^{-2/3} $ again.
Applying the chain rule again, we change the exponent from $-2/3$ to $-5/3$ and adjust the coefficient accordingly.

The calculation provides $ b''(x) = -\frac{2}{9}(-x - 6)^{-5/3} $. Analyzing the sign of $ b''(x) $ tells us whether the function is concave up (positive) or concave down (negative) at different intervals. It's crucial for identifying points of inflection where the concavity changes.

chain rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When a function is made by combining two or more other functions, the chain rule helps find the derivative of the overall combination.

In essence, it states: if a variable $ y = f(g(x)) $, then $ y' = f'(g(x)) \cdot g'(x) $, which means the derivative of $ f $ evaluated at $ g(x) $ is multiplied by the derivative of $ g $.
For $ b(x)=(-x-6)^{1/3} $, the outer function is $ u^{1/3} $ and the inner function is $ u = -x-6 $.
To differentiate using the chain rule, we evaluate the derivative of the outer function first, and then multiply it by the derivative of the inner function.

This two-step process ensures that both components of the composite function are accurately accounted for when differentiating, making the chain rule an indispensable tool in calculus.

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91影视

Determine the interval(s) on which the function is concave up and concave down. $$ b(x)=\sqrt[3]{-x-6} $$

Short Answer

Step by step solution

Understand the function and rewrite it for simplicity

Compute the first derivative \( b'(x) \)

Compute the second derivative \( b''(x) \)

Determine the critical points from the second derivative

Test intervals around critical point to determine concavity

Key Concepts

first derivative

second derivative

chain rule

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