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Plotting a function graph/ 2cosx+15x

Graphing y = 2cosx+15x

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The graph:

from to

Intersection points:

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Piecewise:

The solution

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f(x) = 2*cos(x) + 15*x
f(x)=15x+2cos⁡(x)f{\left(x \right)} = 15 x + 2 \cos{\left(x \right)} 
f = 15*x + 2*cos(x)
The graph of the function
-3.00-2.75-2.50-2.25-2.00-1.75-1.50-1.25-1.00-0.75-0.50-0.250.00-10050
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
15x+2cos⁡(x)=015 x + 2 \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=−0.132170426278227x_{1} = -0.132170426278227
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*cos(x) + 15*x.
0⋅15+2cos⁡(0)0 \cdot 15 + 2 \cos{\left(0 \right)}
The result:
f(0)=2f{\left(0 \right)} = 2
The point:
(0, 2)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
15−2sin⁡(x)=015 - 2 \sin{\left(x \right)} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−2cos⁡(x)=0- 2 \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Convex at the intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(15x+2cos⁡(x))=−∞\lim_{x \to -\infty}\left(15 x + 2 \cos{\left(x \right)}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(15x+2cos⁡(x))=∞\lim_{x \to \infty}\left(15 x + 2 \cos{\left(x \right)}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*cos(x) + 15*x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(15x+2cos⁡(x)x)=15\lim_{x \to -\infty}\left(\frac{15 x + 2 \cos{\left(x \right)}}{x}\right) = 15
Let's take the limit
so,
inclined asymptote equation on the left:
y=15xy = 15 x
lim⁡x→∞(15x+2cos⁡(x)x)=15\lim_{x \to \infty}\left(\frac{15 x + 2 \cos{\left(x \right)}}{x}\right) = 15
Let's take the limit
so,
inclined asymptote equation on the right:
y=15xy = 15 x
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
15x+2cos⁡(x)=−15x+2cos⁡(x)15 x + 2 \cos{\left(x \right)} = - 15 x + 2 \cos{\left(x \right)}
- No
15x+2cos⁡(x)=15x−2cos⁡(x)15 x + 2 \cos{\left(x \right)} = 15 x - 2 \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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