f(x)=x⁴+5x²-4 x intercept and y intercept number of turning point sketch of graph
Answers & Comments
raven2003
In the graph y= ((x^2) /2) (x+1) (3-x) (2-x), is 0,0 an inflection point or a turning point? This can be answered without the tedious work of finding the first and second derivatives.
Note that the graph of y will have roots at -1, 2, and 3 of multiplicity 1, so the graph passes through the x-axis at those values, and a root at 0 of multiplicity two. If a root has an even multiplicity, then the graph is tangent to the x-axis at that point but does not pass through.
The lead coefficient of the quintic expression is positive, so it goes to -inf as x goes to -inf and to +inf as x goes to +inf..
Put it all together and you must have a local minimum at (0,0), not an inflection point
Answers & Comments
This can be answered without the tedious work of finding the first and second derivatives.
Note that the graph of y will have roots at -1, 2, and 3 of multiplicity 1, so the graph passes through the x-axis at those values, and a root at 0 of multiplicity two. If a root has an even multiplicity, then the graph is tangent to the x-axis at that point but does not pass through.
The lead coefficient of the quintic expression is positive, so it goes to -inf as x goes to -inf and to +inf as x goes to +inf..
Put it all together and you must have a local minimum at (0,0), not an inflection point