For f(x)=2(x+5)^3 +7
Find and classify the extreme values, determine where the function is increasing and decreasing, where it is concave up and concave down, and any points of inflection.
I understand this is using the first and second derivative tests, but I can't find any zeroes for the derivative of this function. Am I doing something wrong?
Think of the graph for y = x^3
It has no max/min, but just an inflection point at x=0.
You function is just the same, only stretched by 2, shifted left by 5, and shifted up by 7.
f(x)=0=2(x+5)^3 +7
(x+5)^3=-7/2
x+5= -cubroot(7/2)
x=-5-cubroot(7/2)
Those are the zeroes, but I don't see in the problem wording it asked for a zero
y = 2 (x+5)^3 + 7
to make this really easy you can say
z = x+5
then dz/dx = 1
y = 2 z^3 + 7
dy/dz = 6 z^2
d^2y/dz^2 = 12 z
so
dy/dx = dy/dz * dz/dx = 6(x^2+10x+25)
when is that zero ?
(x+5)(x+5) = 0
x = -5 twice
what is d/dx (dy/dx)?
6 ( 2 x + 10)
when x = -5
2(-5)+10 = 0
so that is not a max or a min but an inflection point.
To find the extreme values, increasing and decreasing intervals, and the concavity of the function, we need to find the first and second derivatives of the given function. Let's start by finding the first derivative.
Step 1: Find the first derivative, f'(x):
f(x) = 2(x+5)^3 + 7
Using the power rule and chain rule, we have:
f'(x) = 2 * 3(x+5)^2 * 1
Simplifying this expression, we get:
f'(x) = 6(x+5)^2
Step 2: Now, let's find the second derivative, f''(x):
To find the second derivative, we differentiate the first derivative with respect to x.
f''(x) = 2 * 2 * (x+5)^1 * 1
Simplifying this expression, we get:
f''(x) = 12(x+5)
Now, let's address your concern about not being able to find any zeroes for the derivative of this function.
In the given function, f(x) = 2(x+5)^3 + 7, the first derivative, f'(x) = 6(x+5)^2, is a perfect square. This means that the first derivative does not have any zeros, and consequently, there are no critical points where the function may have local extreme values or change its concavity.
However, we can still determine where the function is increasing or decreasing and its concave up or concave down by analyzing the signs of the first and second derivatives.
1. Increasing and Decreasing:
To determine where the function is increasing or decreasing, we examine the sign of the first derivative, f'(x).
When f'(x) > 0, the function is increasing.
When f'(x) < 0, the function is decreasing.
Since f'(x) = 6(x+5)^2 is positive everywhere except when x = -5, the function is increasing for all x ≠ -5.
2. Concavity:
To determine the concavity of the function, we examine the sign of the second derivative, f''(x).
When f''(x) > 0, the function is concave up.
When f''(x) < 0, the function is concave down.
Since f''(x) = 12(x+5) is positive for all x ≠ -5, the function is concave up everywhere except at x = -5.
3. Points of Inflection:
Points of inflection occur where the concavity changes. Since the second derivative is always positive, there are no points of inflection for this function.
In summary:
- The function is always increasing.
- The function is always concave up.
- There are no points of inflection.