Check if the function f(x)= 10x^2 - 2x + 5 as extreme values.

The given function has a single extreme value, namely a
minimum value (bcause the coefficient of x^2 is positive). The minimum value is
represented by the vertex of the parable, whose expression is 10x^2 - 2x +
5.

The coordinates of the vertex are: V(xV ,
yV):

xV = -b/2a

yV =
-delta/4a

delta = b^2 -
4ac

We'll identify the coefficients
a,b,c:

a = 10

b =
-2

c = 5

Now, we'll determine
the coordinates of the vertex:

xV =
-(-2)/2*10

xV =
1/10

xV =
0.1

yV = -(4 -
200)/4*10

yV = 196/4*10

yV =
49/10

yV =
4.9

The function has just one
extreme point and it's coordinates are: (0.1 ,
4.9).

Another manner to verify
the existence and the number of extreme points of a function is to differentiate the
function.

We'll differentiate
f(x).

f'(x) = (10x^2 - 2x +
5)'

f'(x) = 20x - 2

We'll
calculate the roots of f'(x):

f'(x) =
0

20x - 2 = 0

We'll divide by
2:

10x - 1 = 0

x =
1/10

The function has an extreme point for any root of the
derivative.

Since the derivative has just one root, the
function will have just a single extreme point.

f(1/10) =
10/100 - 2/10 + 5

f(1/10) = 1/10 - 2/10 +
5

f(1/10) = -1/10 + 5

f(1/10)
= (-1+50)/10

f(1/10) = 49/10