5.3 Determining Intervals on Which a Function is Increasing or Decreasing

Cards (56)

To find critical points, set $ f'(x) = 0 $ and solve for x
Why are critical points important in analyzing functions?
Identify increasing or decreasing intervals
Match the sign of $ f'(x) $ with the behavior of $ f(x) $:
+$ $f'(x) > 0$ $ ↔️ Increasing
-$ $f'(x) < 0$ $ ↔️ Decreasing
0 ↔️ Constant
The first derivative of $ f(x) = x^3 - 3x $ is defined for all values of $ x $. 
True
If $ f'(x) > 0 $ in an interval, then $ f(x) $ is increasing
For the function $ f(x) = x^3 - 6x^2 + 5 $, the first derivative is $ f'(x) = 3x^2 - 12x $. The critical points are $ x = 0 $ and $ x = 4 $. At $ x = -1 $, $ f'(-1) = 15 > 0 $, so $ f(x) $ is increasing
What is the first key step in the First Derivative Test?
Identify critical points
If $ f'(x) > 0 $, then $ f(x) $ is increasing. 
True
The critical points of a function occur where $ f'(x) = 0 $ or $ f'(x) $ is undefined
If $ f'(x) < 0 $, the function $ f(x) $ is decreasing. 
True
Steps to determine intervals where a function is increasing using the First Derivative Test
1️⃣ Identify the critical points
2️⃣ Choose test values within intervals defined by critical points
3️⃣ Evaluate $ f'(x) $ at the test values
4️⃣ Analyze the sign of $ f'(x) $ to determine behavior
What is the first step to identify critical points of a function?
Find $ f'(x) $
What is the first step in finding the critical points of a function $f(x)$?
Set $f'(x) = 0$
What is the derivative of $f(x) = x^3 - 3x$?
$f'(x) = 3x^2 - 3$
What does analyzing the sign of $f'(x)$ help determine about $f(x)$?
Its increasing, decreasing, or constant behavior
On what interval is $f(x) = x^3 - 3x$ increasing?
$(-\infty, -1)$ and $(1, \infty)$
What is the value of $f'(2)$ for $f(x) = x^3 - 3x$?
$f'(2) = 9$
The critical points of $f(x) = x^3 - 3x$ are $x = -1$ and $x = 1$.
True
Points where $ f'(x) $ is undefined must also be considered as critical points. 
True
If $ f'(x) > 0 $ in an interval, then $ f(x) $ is increasing
What is the first derivative of $ f(x) = x^3 - 6x^2 + 5 $?
$f'(x) =$ $3x^{2} - 12x$
The critical points of $ f(x) = x^3 - 6x^2 + 5 $ are $ x = 0 $ and \( x = 4
What are the critical points of $ f(x) = x^3 - 3x $?
$x =$ $\pm 1$
If $ f'(x) < 0 $ in an interval, then $ f(x) $ is decreasing. 
True
Steps to analyze the sign of the first derivative
1️⃣ Find the first derivative $ f'(x) $
2️⃣ Solve $ f'(x) = 0 $ to find critical points
3️⃣ Choose test values within each interval
4️⃣ Evaluate $ f'(x) $ at the test values
5️⃣ Analyze the sign of $ f'(x) $ to determine function behavior
In the First Derivative Test, we evaluate $ f'(x) $ at each test value to determine its sign.
What are the critical points of $ f(x) = x^3 - 3x $?
$ x = -1 $ and $ x = 1 $
What does a positive value of $ f'(x) $ indicate about the behavior of $ f(x) $?
Increasing
What is the behavior of $ f(x) $ when $ f'(x) = 0 $?
Constant
The function $ f(x) = x^3 - 6x^2 + 5 $ is decreasing on the interval $ (0, 4) $. 
True
For the function $ f(x) = x^3 - 3x $, the critical points are \( x = \pm 1
The derivative $f'(x)$ can be undefined at points where the function has vertical tangents or discontinuities. 
True
Solving $3x^2 - 3 = 0$ gives critical points $x = \pm $1
Match the sign of $f'(x)$ with the behavior of $f(x)$:
$f'(x) > 0$ ↔️ Increasing
$f'(x) < 0$ ↔️ Decreasing
$f'(x) = 0$ ↔️ Constant
The First Derivative Test helps determine where a function is increasing or decreasing based on the sign of $f'(x)$. 
True
When $x = 0$, $f'(x) = -3$, which means $f(x)$ is decreasing
When $f'(x) > 0$, the function $f(x)$ is increasing
What is a critical point of a function?
f'(x) = 0 or undefined
Steps to find critical points
1️⃣ Calculate $ f'(x) $
2️⃣ Solve $ f'(x) = 0 $
3️⃣ Check for undefined $ f'(x) $
4️⃣ List all critical points
If $ f'(x) < 0 $ in an interval, then $ f(x) $ is decreasing.
True