f(x)=(5x-4)e^2x
How do you find the one critical number??
Differentiate and put equal to zero.
f'[x] = 5e^(2x) + 2e^(2x)(-4+5x)
f'[0] = -3
So the x-coordinate is -3
I'd guess that that is what the question wants.
Hope that helps
I made a slight error. i don't want f'[0], i want f'[x] = 0 to find x = something.
so we get 5e^(2x) + 2e^(2x)(-4+5x) = 0
divide across by e^(2x)
we get 5 + 2(-4+5x) = 0
5 - 8 + 10x =0
10x = 3
x = 3/10
Sorry about the mistake in my first post.
hope that helps
To find the critical number of a function, you need to locate the values of x where the derivative of the function is equal to zero or undefined. In this case, we need to find the critical number of the function f(x) = (5x - 4)e^(2x).
To do this, we will follow these steps:
Step 1: Find the derivative of the function f(x) with respect to x.
Step 2: Set the derived function equal to zero and solve for x.
Step 3: Check if the derived function is undefined for any value of x.
Let's go through the process step by step.
Step 1: Find the derivative of f(x):
To find the derivative of f(x), we will use the product rule and the chain rule. The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
Using the product rule, the derivative of f(x) = (5x - 4)e^(2x) is:
f'(x) = (5x - 4) * (d/dx)(e^(2x)) + e^(2x) * (d/dx)(5x - 4)
To find the derivative of e^(2x), we apply the chain rule. The derivative of e^u, where u is a function of x, is given by:
(d/dx)(e^u) = e^u * (d/du)(u)
Therefore, the derivative of f(x) becomes:
f'(x) = (5x - 4) * 2e^(2x) + e^(2x) * 5
Simplifying further, we get:
f'(x) = 10xe^(2x) + e^(2x) - 4e^(2x)
Step 2: Set f'(x) equal to zero and solve for x:
To find the critical number, we set the derivative f'(x) equal to zero and solve for x:
10xe^(2x) + e^(2x) - 4e^(2x) = 0
Let's factor out e^(2x):
e^(2x) * (10x + 1 - 4) = 0
Simplifying further, we get:
e^(2x) * (10x - 3) = 0
To find the critical number, we need to solve the equation 10x - 3 = 0:
10x - 3 = 0
10x = 3
x = 3/10
Therefore, the critical number of the function f(x) = (5x - 4)e^(2x) is x = 3/10.