The function h(x) = x^3+bx+d has a critical point at (2,-4). Determine the constants b and d and find the equation of h(x). I'm very confused and I do not have any idea on what the first step is.
To determine the constants b and d and find the equation of h(x), we can use the fact that the given function has a critical point at (2, -4).
In order to find the equation of h(x), we need to use the given information to determine the values of b and d.
Step 1: Understanding Critical Points
A critical point is a point on the graph of a function where either the derivative is zero or does not exist. In this case, since the function h(x) = x^3 + bx + d, the derivative, h'(x), would be the slope of h(x) at any given point x. For a critical point (c, h(c)), the derivative at that point is zero, i.e., h'(c) = 0.
Step 2: Finding the Derivative of h(x)
To find the derivative of h(x), we need to differentiate it with respect to x. In this case, h(x) = x^3 + bx + d, so we can find h'(x) as follows:
h'(x) = 3x^2 + b
Step 3: Evaluating the Derivative at the Critical Point
Since h(x) has a critical point at (2, -4), we can substitute x = 2 into the derivative h'(x) and set it equal to zero:
h'(2) = 3(2)^2 + b = 0
Step 4: Solving for b
Using the equation 3(2)^2 + b = 0, we can solve for b as follows:
12 + b = 0
b = -12
Step 5: Determining the Value of d
In order to determine the value of d, we substitute the x and y coordinates of the critical point (2, -4) into the original function h(x). Thus:
h(2) = (2)^3 + (-12)(2) + d = -4
Simplifying this equation, we get:
8 - 24 + d = -4
-16 + d = -4
d = 12
Step 6: Writing the Equation of h(x)
By substituting the values of b and d we found into the original function h(x), we can write the equation of h(x) as follows:
h(x) = x^3 - 12x + 12