NOTE: When using interval notation in WeBWork, remember that: You use 'INF' for o and '-INF' for -o. And use 'U' for the union symbol. Enter DNE if an answer does not exist. f(x) = 5 sin x +4 cos x, on (- π, π) a) Determine the intervals on which f is concave up and concave down.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
NEED HELP WITH CALCULUS. I TRIED THIS N GOT IT WRONG. PLEASE DONT USE "TAN" WORD IN YOUR ANSWER. WRITE THE ANS IN TERMS OF "PI". AND DONT MAKE IT TOO COMPLEX.
![### Calculus: Analyzing Concavity and Extrema
#### Problem Statement
Given the function \( f(x) = 5 \sin x + 4 \cos x \), defined on the interval \((-π, π)\):
### Part (a): Determine the Intervals of Concavity
1. **Concave Up:**
\( f \) is concave up on:
\[
\left( -\pi, -\tan^{-1}\left(\frac{40}{9}/2\right) \right) \cup \left( \pi (-\tan^{-1}\left(\frac{40}{9}/2\right) \right)
\]
2. **Concave Down:**
\( f \) is concave down on:
\[
(-\pi, 2.4669)
\]
### Part (b): Inflection Points
Based on the answer to part (a), the inflection points of \( f \) are:
\[
(2.4669, 0)
\]
*Note: Enter multiple answers separated by commas.*
### Part (c): Critical Numbers and Relative Extrema
Find the critical numbers of \( f \) and use the Second Derivative Test to determine the relative extrema.
1. **Relative Maxima:**
Relative maxima occur at:
\[
0.897
\]
2. **Relative Minima:**
Relative minima occur at:
\[
\text{dne}
\]
### Part (d): \( x \)-Values for Relative Maxima or Minima of \( f' \)
Determine the \( x \)-values where \( f'(x) \) has a relative maximum or minimum.
1. **Relative Maxima of \( f' \):**
\( f' \) has a relative maximum at:
\[
(2.4669)
\]
2. **Relative Minima of \( f' \):**
\( f' \) has a relative minimum at:
\[
0
\]
*Note: Enter multiple answers separated by commas.*
This exercise helps in understanding the concavity, inflection points, and relative extrema of trigonometric functions using calculus concepts such as the second derivative test.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66996606-3408-45a2-8bcb-4b1c908314bf%2Fab505b2e-5aa0-4659-83cc-9ca39c5e83eb%2Fx5r5ht_processed.png&w=3840&q=75)
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