The graph y = f(x) = x/(x⁴-16) (x divide by (x to the power of 4 minus 16)) - plot the function graph and draw it. Curve sketching this function. [THERE'S THE ANSWER!] online
The points at which the function is not precisely defined: x1=−2 x2=2
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: x4−16x=0 Solve this equation The points of intersection with the axis X:
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0: substitute x = 0 to x/(x^4 - 16). −16+040 The result: f(0)=0 The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation dxdf(x)=0 (the derivative equals zero), and the roots of this equation are the extrema of this function: dxdf(x)= the first derivative −(x4−16)24x4+x4−161=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 dx2d2f(x)=0 (the second derivative equals zero), the roots of this equation will be the inflection points for the specified function graph: dx2d2f(x)= the second derivative (x4−16)24x3(x4−168x4−5)=0 Solve this equation The roots of this equation x1=0 You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function: Points where there is an indetermination: x1=−2 x2=2
x→−2−lim(x4−16)24x3(x4−168x4−5)=−∞ x→−2+lim(x4−16)24x3(x4−168x4−5)=∞ - the limits are not equal, so x1=−2 - is an inflection point x→2−lim(x4−16)24x3(x4−168x4−5)=−∞ x→2+lim(x4−16)24x3(x4−168x4−5)=∞ - the limits are not equal, so x2=2 - is an inflection point
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: Have no bends at the whole real axis
Vertical asymptotes
Have: x1=−2 x2=2
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo x→−∞lim(x4−16x)=0 Let's take the limit so, equation of the horizontal asymptote on the left: y=0 x→∞lim(x4−16x)=0 Let's take the limit so, equation of the horizontal asymptote on the right: y=0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x/(x^4 - 16), divided by x at x->+oo and x ->-oo x→−∞limx4−161=0 Let's take the limit so, inclined coincides with the horizontal asymptote on the right x→∞limx4−161=0 Let's take the limit so, inclined coincides with the horizontal asymptote on the left
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: x4−16x=−x4−16x - No x4−16x=x4−16x - Yes so, the function is odd