5. Graphical Applications of Derivatives

The First Derivative Test

5. Graphical Applications of Derivatives

The First Derivative Test

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Identify the intervals on which the function is increasing or decreasing.
$f(x)=\sin^2x$ on $[0,\pi]$

Increasing on $[0,π]$

Decreasing on $[0,π]$

Increasing on $[0,\frac{\pi}{2})$ , Decreasing on $(\frac{\pi}{2},\pi]$

Increasing on $(\frac{\pi}{2},\pi]$ , Decreasing on $[0,\frac{\pi}{2})$

Verified step by step guidance

First, understand that to determine where the function f(x) = sin^2(x) is increasing or decreasing, we need to find its derivative, f'(x). This will help us identify the critical points and intervals of increase or decrease.

Calculate the derivative of f(x) = sin^2(x). Use the chain rule: if f(x) = (sin(x))^2, then f'(x) = 2 * sin(x) * cos(x). This simplifies to f'(x) = sin(2x) using the double angle identity.

Set the derivative f'(x) = sin(2x) equal to zero to find the critical points. Solve sin(2x) = 0 for x in the interval [0, π]. The solutions are x = 0, π/2, and π.

Analyze the sign of f'(x) = sin(2x) in the intervals determined by the critical points: [0, π/2), (π/2, π]. If f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing.

Conclude the intervals: f(x) is increasing on [0, π/2) because sin(2x) > 0, and decreasing on (π/2, π] because sin(2x) < 0.

7:32m

Watch next

Master Determining Where a Function is Increasing & Decreasing with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Sketch the graph of a twice-differentiable function y=f(x) that passes through the points (-2,2), (-1,1), (0,0),(1,1), and (2,2) and whose first two derivatives have the following sign patterns.

Textbook Question

In Exercises 121–124, find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function’s first and second derivatives. How are the values at which these graphs intersect the x-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function?

123. y=(4/5)x^5+16x^2-25

Multiple Choice

Identify the open intervals on which the function is increasing or decreasing.

$f(x)=3x^4+8x^3-18x^2+7$

Multiple Choice

Identify the open intervals on which the function is increasing or decreasing.

$f(x)=x^{2/3}(4-x)$

Multiple Choice

Identify the local minimum and maximum values of the given function, if any.

$f\left(\theta\right)=\sin\theta+\cos^2\theta$ on $[0,π]$

Textbook Question

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x² + 3 on [-3,2]

Textbook Question

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x√(4 - x²) on [-2,2]

Textbook Question

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = 2x⁵ - 5x⁴ - 10x³ + 4 on [-2,4]

Show more practices

The First Derivative Test practice set

Problem sets built by lead tutors

Expert video explanations

Go to Practice

Struggling with Calculus?

Identify the intervals on which the function is increasing or decreasing.f(x)=sin2xf(x)=\sin^2xf(x)=sin2x on [0,π][0,\pi][0,π]

Watch next

The First Derivative Test practice set

Identify the intervals on which the function is increasing or decreasing.
$f(x)=\sin^2x$ on $[0,\pi]$