Explained: "How do you find all points on the..."

High School
Calculus and Analysis
Calculus 1
Derivatives

Magdalena Warren

Answered question

2022-04-17

How do you find all points on the graph of the function

f (x) = 2 \sin x + \sin^{2} x

at which the tangent line is horizontal?

Answer & Explanation

vadomosigmx6

Beginner2022-04-18Added 12 answers

Explanation:
We know that the derivative represents the instantaneous rate of change of a function, or the slope of the function. We also know that the slope of a line that is horizontal is 0. We start by differentiating the function.

f (x) = 2 \sin x + \sin^{2} x

f (x) = 2 \sin x + (\sin x) (\sin x)

f^{'} (x) = (0 \times \sin x + 2 \times \cos x) + (\cos x \sin x + \cos x \sin x)

f^{'} (x) = 2 \cos x + 2 \cos x \sin x

We set this to 0 an solve for x.

0 = 2 \cos x + 2 \cos x \sin x

0 = 2 (\cos x + \cos x \sin x)

0 = \cos x (1 + \sin x)

\cos x = 0

and

\sin x = - 1

x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function
$f (x, y) = x^{3} - 6 x y + 8 y^{3}$
$\frac{1}{\sec (x)}$ in derivative?
What is the derivative of $\ln (x + 1)$ ?
What is the limit of $e^{- x}$ as $x \to \infty$ ?
What is the derivative of $f (x) = 5^{\ln x}$ ?
What is the derivative of $e^{- 2 x}$ ?
How to find $lim \frac{e^{t} - 1}{t}$ as $t \to 0$ using l'Hospital's Rule?
What is the integral of $\sqrt{9 - x^{2}}$ ?
What is the derivative of $f (x) = \ln [x^{9} {(x + 3)}^{6} {(x^{2} + 7)}^{5}]$ ?
What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?
How to find the derivative of $y = e^{5 x}$ ?
How to evaluate the limit $\frac{\sin (5 x)}{x}$ as x approaches 0?
How to find derivatives of parametric functions?
What is the antiderivative of $- 5 e^{x - 1}$ ?
How to evaluate: indefinite integral $\frac{1 + x}{1 + x^{2}} d x$ ?

Didn't find what you were looking for?