Question

The graph of the polynomial p(x) = 2x^2 + 4x + 3 has its least value at the point.

Answer 1

Step-by-step explanation:

f(x) = 2 x² + 4x +3

f'(x)= 4x+4

f"(x)=4

To find critical number, f'(x)=0

4x+4=0

x= -1

66

when x=-1, f"(x) is positive

By second derivative test, f atttains minimum at x= -1.

The minimum value=f(-1)

=2-4+3=1

The minimum point is (-1,1)

Answer 2

Given : -

• Polynomial function is p(x) = 2x² + 4x + 3

To find : -

• The coordinates where the given polynomial has its least value.

Solution : -

We are aware about the general form of a quadratic polynomial :

• ax² + bx + c

By comparing it with the given polynomial, we get :

• a = 2
• b = 4
• c = 3

To find the least value of x, we use the formula :

• Min ( x ) = -b/2a

So,

⇒ Min ( x ) = -4/(2×2)

⇒ Min ( x ) = - 4/4

⇒ Min ( x ) = -1

So the value of x where the polynomial has it's least value is -1.

Now substitute value of x in given polynomial to find y

⇒ p(x) = 2x² + 4x + 3

⇒ p(-1) = 2 (-1)² + 4(-1) + 3

⇒ y = 2 ( 1 ) - 4 + 3

⇒ y = 2 - 4 + 3

⇒ y = 1

From our above results, we obtained that, the minimum value of the given polynomial function is on (-1,1)

Refer to the graph in attachment for verification.