Problem 49.
Find the critical values
f(x) = (4x^2 - 9) / x
before differentiating, I would simplify it to
f(x) = 4x - 9/x
then f'(x) = 4 + 9/(x^2)
set that equal to zero, and solve to find any maximum and minimum values
f"(x) = -18/(x^3)
when you set the second derivative equal to zero, there is no solution, so there is no point of inflection.
Use your second derivative to determine if the curve is concave up or down.
If f" is positive, the graph is concave up
etc. you should know these properties to do these kind of questions.
Thanks.
To find the critical values of a function, we need to determine the values of x for which the derivative of the function is equal to zero or is undefined.
Let's start by finding the derivative of the given function f(x) = (4x^2 - 9) / x.
Step 1: Apply the quotient rule.
Applying the quotient rule, the derivative of f(x) is given by:
f'(x) = (x * d/dx(4x^2 - 9) - (4x^2 - 9) * d/dx(x)) / (x^2)
Step 2: Simplify the expression.
Simplifying the expression, we get:
f'(x) = (8x^2 - 9 - (4x^2 - 9)) / (x^2) = (8x^2 - 9 - 4x^2 + 9) / (x^2) = (4x^2) / (x^2) = 4
Step 3: Set the derivative equal to zero.
To find the critical values, we need to set the derivative equal to zero and solve for x:
4 = 0
Since the equation 4 = 0 has no solution, there are no critical values for the function f(x).
Therefore, the function f(x) = (4x^2 - 9) / x has no critical values.