How many local maxima and minima does the polynomial y=x^4-9x^2+7 have?

To determine the local extrema of a function, we'll have
to find out first if the function does have any critical
values.

The critical values are the roots of the first
derivative. We'll differentiate the given function with respect to x, to determine the
1st derivative of the function.

f'(x) = 4x^3 -
18x

Now, we'll cancel the 1st derivative to determine it's
zeroes:

4x^3 - 18x = 0

2x(2x^2
- 9) = 0

We'll cancel each
factor:

2x = 0 => x1 =
0

2x^2 - 9 = 0 => 2x^2 =
9

x^2 = 9/2

x2 =
3/sqrt2

x2 = 3sqrt2/2

x3 =
-3sqrt2/2

We'll determine the monotony of the function over
the intervals (-`oo`; -3sqrt2/2 ) ; (-3sqrt2/2 ; 0) ; (0;3sqrt2/2) ; (3sqrt2/2 ; `oo`
).

The function is decreasing over the interval (-`oo` ;
-3sqrt2/2) and then it increases over (-3sqrt2/2 ; 0), therefore the point f(-3sqrt2/2)
represents a minimum local point.

The function is
increasing over the interval ( -3sqrt2/2 ; 0) and then it decreases over (0 ; 3sqrt2/2
), therefore the point f(0) represents a maximum local
point.

The function is decreasing over the interval (0 ;
3sqrt2/2 ) and then it increases over (3sqrt2/2 ; `oo` ), therefore the point
f(3sqrt2/2) represents a minimum local
point.

Therefore, the local minima of the
function are represented by the values f(-3sqrt2/2) and f(3sqrt2/2) and the local maxima
is represented by the value f(0).