Question

If f(x)=ax^(2)+b, f(−3)=4 and f′(−3)=4, find the coefficients a and b.

a =
b =

Answer 1

f' = 2ax

so,
9a+b = 4
-6a = 4

f(x) = -2/3 x^2 + 10

Answer 2

from your f(x)

f'(x) = 2ax
given: f'(-3) = 4
4 = 2a(-3)
a = - 2/3

also f(-3) = 4
in original:
4 = (-2/3)(-3)^2 + b
4 + 6 = b
b = 10

check my arithmetic

Answer 3

f₍x₎ = a x² + b

f '₍x₎ = 2 a x

f(- 3) = 4 means:

x = - 3 , f₍- 3₎ = 4

a x² + b = 4

a ∙ ( - 3 )² + b = 4

a ∙ 9 + b = 4

9 a + b = 4

f′(- 3) = 4 means

x = - 3 , f '₍- 3₎ = 2 a x

2 a x = 4

2 a ∙ ( - 3 ) = 4

- 6 a = 4

Divide both sides by - 6

a = - 4 / 6 =

a = 2 ∙ ( - 2 ) / 2 ∙ 3

a = - 2 / 3

Put this value in equation:

9 a + b = 4

9 ∙ ( - 2 / 3 ) + b = 4

- 18 / 3 + b = 4

- 6 + b = 4

Add 6 to both sides

b = 10

The solutions are:

a = - 2 / 3 , b = 10

f₍x₎ = - 2 / 3 x² +10

Answer 4

[AI-generated answer — Explain Bot]

To find the coefficients a and b, we can use the given information and the properties of a derivative.

First, let's find the equation for the derivative of f(x), f'(x). Since f(x) = ax^2 + b, we can find the derivative by taking the derivative of each term separately. The derivative of ax^2 is 2ax (using the power rule), and the derivative of b is 0 (since b is a constant). So, f'(x) = 2ax.

Now, we are given that f(-3) = 4. Using this information, we can substitute x = -3 into the original function f(x) and solve for a and b.

f(-3) = a(-3)^2 + b
4 = 9a + b (equation 1)

Next, we are given that f'(-3) = 4. Using this information, we can substitute x = -3 into the derivative f'(x) = 2ax and solve for a.

f'(-3) = 2a(-3)
4 = -6a
a = -4/6
a = -2/3 (equation 2)

Now, we can substitute the value of a from equation 2 into equation 1 and solve for b.

4 = 9(-2/3) + b
4 = -6 + b
b = 10

Therefore, the coefficients a and b are:

a = -2/3
b = 10