What is the value of b and the value of c.

Question

Given that the graph of the function h(x) = 3x³ + bx² + cx + 20 is tangent to the x-axis at x = 2 , find the value of b and the value of c.

Answer 1

h'(2) = 0 since graph is tangent to x axis at x = 2 and the x axis has slope = 0.

h(2) = 0 as well since y value of X axis at x=2 is 0

 

you will have 2 equations and 2 unknowns

h(2)= 3(2)³ + b(2)²+c(2)+20 = 0

thus

24+4b+2c+20 = 0

or

4b+2c=-44 (*)

 

h'(2) = 3*3(2)²+2b(2)+c= 0

thus

36+4b+c = 0

or

4b +c = -36 (**)

 

2 equations 2 unknowns you can solve.

you will get c =-8 and b = =-7.

 

hope helpful.

best - David  

Answer 2

Since the graph is tangent to the x-axis at x=2,

 

h'(2) = 0  ---> This indicates the derivative

 

h'(x) = 9x² + 2bx + c

 

 

Therefore,

 

9(2)² + 2b(2) + c = 0

36 + 4b + c = 0

        4b + c = -36                     eq1

 

h(2) = 0     ---->  point on the x-axis

 

3(2)³ + b(2)² + c(2) + 20 = 0

24 + 4b + 2c + 20 = 0

        4b + 2c = -44                   eq2

 

 

The bolded equations are your system of equations.  Use the substitution/elimination methods to solve for b and c.

Answer 3

Hi Barbra, when the function is tangent to the x-axis, the y value is 0. Solving for b and c follows from substituting for x = 2, and h(x) = 0.

 

3x³ + bx² + cx + 20 = h(x)

 

3 * 2³ + b * 2² + c * 2 + 20 = 0

 

24 + 4b + 2c + 20 = 0

 

4b + 2c + 44 = 0

 

Subtract 44 from both sides:

 

4b + 2c = -44

 

Divide all terms by 2:

 

2b + c = -22

 

Subtract 2b from both sides:

 

c = -2b - 22

 

Let b be any number, then c can be found.

 

Example:

 

Let b = 1

 

c = (-2 * 1) - 22

 

c = -24

 

Substitute for b and c in the original equation:

 

3 * x³ + 1 * x² - 24 * x + 20 = h(x)

 

When x = 2, y = 0:

 

24 + 4 - 48 + 20 = 0

 

0 = 0

 

Questions?