Question 558544
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
{{{graph(300,300,-1.5,1.5,-2,14,5x^4-7x^2+4)}}} 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
{{{f(x)=5(x^2-7/10)^2+31/20}}}
which tells you that {{{f(x)>=31/20}}}.
The minima will occur when {{{x^2=7/10}}} and you will have {{{f(x)=31/20}}} then.
You could also try to solve
{{{5x^4-7x^2+4=0}}} by changing variables with {{{y=x^2}}} to find that there are no real solutions for y, so there are no rela solutions for x either.