Suppose the function f(x)=2x^3+ax^2+bx has a critical point at x=−1, for which f(−1)=−2. What is a, b?

To find the values of a and b, we need to use the properties of critical points. A critical point occurs when the derivative of the function is equal to zero.

First, let's find the derivative of the function f(x):
f'(x) = 6x^2 + 2ax + b

Since the critical point occurs at x = -1, we can substitute this value into the derivative:
f'(-1) = 6(-1)^2 + 2a(-1) + b
= 6 + (-2a) + b
= 6 - 2a + b

We know that the derivative at the critical point is zero, so:
f'(-1) = 0

Substituting this into the equation, we have:
0 = 6 - 2a + b

Next, we know that f(-1) = -2. Plugging this into the original function, we have:
f(-1) = 2(-1)^3 + a(-1)^2 + b
= -2 + a + b

We can now set up a system of equations using the two equations we obtained:
0 = 6 - 2a + b (Equation 1)
-2 = -2 + a + b (Equation 2)

Simplifying Equation 2, we get:
a + b = 0 (Equation 3)

Now we can solve the system of equations (Equations 1 and 3):
From Equation 3, we have b = -a.
Substituting this into Equation 1, we get:
0 = 6 - 2a - a
3a = 6
a = 2

Finally, substituting the value of a into Equation 3, we have:
2 + b = 0
b = -2

Therefore, the values of a and b are a = 2 and b = -2.

To find the values of a and b, we can use the properties of critical points and the given information.

A critical point occurs where the derivative of the function is equal to zero. Therefore, we need to find the derivative of f(x) and set it equal to zero to solve for the values of a and b.

Let's start by finding the derivative of f(x):

f(x) = 2x^3 + ax^2 + bx

Taking the derivative of each term:

f'(x) = d/dx (2x^3) + d/dx (ax^2) + d/dx (bx)

Using the power rule for differentiation:

f'(x) = 6x^2 + 2ax + b

Since we now know that the critical point is at x = -1, we can substitute this value into the derivative equation to solve for a and b:

f'(-1) = 6(-1)^2 + 2a(-1) + b

Simplifying,

f'(-1) = 6 + (-2a) + b

Now, we know that at the critical point x = -1, f(-1) = -2. Therefore, we can substitute this information into the original function f(x) to find the value of f(-1):

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

Simplifying,

f(-1) = -2 + a + b

Since we know that f(-1) = -2, we can equate this to the known value:

-2 + a + b = -2

From this equation, we can solve for a and b.

By comparing the coefficients of a and b in the two equations, we can set up a system of equations:

-2a + b = 6
a + b = 0

Now, we can solve this system of equations. By subtracting the second equation from the first equation, we get:

-3a = 6

Dividing both sides by -3, we find:

a = -2

Substituting the value of a back into the second equation, we can solve for b:

-2 + b = 0

Adding 2 to both sides, we find:

b = 2

Therefore, the values of a and b that satisfy the given conditions are a = -2 and b = 2.