SOLUTION: Let f(x)=5x^4 -7x^2 +4.
a. How many zeros does f(x) have?
b. Based on the graph, how many real number zeros does f(x) have?
I know the answers are a. 4 and b.2 but I don't
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Question 558544: Let f(x)=5x^4 -7x^2 +4.
a. How many zeros does f(x) have?
b. Based on the graph, how many real number zeros does f(x) have?
I know the answers are a. 4 and b.2 but I don't understand how you get that. Can you explain please? These are from my exam review packet. Help !!
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
a. I agree that it should have 4 complex zeros, because it is a polynomial of degree 4.
b. Why the reference to a graph? Are you supposed to use a graphing calculator?
Are you supposed to calculate minima and maxima based on derivatives?
A graphing calculator would show you somthing like this
showing that there are no real zeros.
You could graph with just pencil and paper, based on the derivative, if you knew a little calculus. You could analyze the function easily enough.
The function is an even function, symmetrical with respect to the y-axis, meaning that f(x)=f(-x). It is obvious that for x with large absolute value, the function is positive, and it grows without bounds towards both ends (towards -infinity and +infinity).
A little algebra can transform the function into
which tells you that .
The minima will occur when and you will have then.
You could also try to solve
by changing variables with to find that there are no real solutions for y, so there are no rela solutions for x either.
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