Mister Exam
Graphing y = x²-3^x+2
The solution
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2 x f(x) = x - 3 + 2
$$f{\left(x \right)} = \left(- 3^{x} + x^{2}\right) + 2$$
f = -3^x + x^2 + 2
The graph of the function
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:
$$\left(- 3^{x} + x^{2}\right) + 2 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$\left(- 3^{x} + x^{2}\right) + 2 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 1$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- 3^{x} \log{\left(3 \right)} + 2 x = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- 3^{x} \log{\left(3 \right)} + 2 x = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$- 3^{x} \log{\left(3 \right)}^{2} + 2 = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{- 2 \log{\left(\log{\left(3 \right)} \right)} + \log{\left(2 \right)}}{\log{\left(3 \right)}}$$
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
$$\left(-\infty, \frac{- 2 \log{\left(\log{\left(3 \right)} \right)} + \log{\left(2 \right)}}{\log{\left(3 \right)}}\right]$$
Convex at the intervals
$$\left[\frac{- 2 \log{\left(\log{\left(3 \right)} \right)} + \log{\left(2 \right)}}{\log{\left(3 \right)}}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$- 3^{x} \log{\left(3 \right)}^{2} + 2 = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{- 2 \log{\left(\log{\left(3 \right)} \right)} + \log{\left(2 \right)}}{\log{\left(3 \right)}}$$
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
$$\left(-\infty, \frac{- 2 \log{\left(\log{\left(3 \right)} \right)} + \log{\left(2 \right)}}{\log{\left(3 \right)}}\right]$$
Convex at the intervals
$$\left[\frac{- 2 \log{\left(\log{\left(3 \right)} \right)} + \log{\left(2 \right)}}{\log{\left(3 \right)}}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\left(- 3^{x} + x^{2}\right) + 2\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(- 3^{x} + x^{2}\right) + 2\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\left(- 3^{x} + x^{2}\right) + 2\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(- 3^{x} + x^{2}\right) + 2\right) = -\infty$$
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 x^2 - 3^x + 2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(- 3^{x} + x^{2}\right) + 2}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(- 3^{x} + x^{2}\right) + 2}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left(- 3^{x} + x^{2}\right) + 2}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(- 3^{x} + x^{2}\right) + 2}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$\left(- 3^{x} + x^{2}\right) + 2 = x^{2} + 2 - 3^{- x}$$
- No
$$\left(- 3^{x} + x^{2}\right) + 2 = - x^{2} - 2 + 3^{- x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(- 3^{x} + x^{2}\right) + 2 = x^{2} + 2 - 3^{- x}$$
- No
$$\left(- 3^{x} + x^{2}\right) + 2 = - x^{2} - 2 + 3^{- x}$$
- No
so, the function
not is
neither even, nor odd