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Question

let F(x) = x^3+3x^2-24x+5

Determine the subintervals of increase and decrease of f
Find local extreme (Local max and min) of f
Determine the concavity of f and the point of inflection

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Answer 1

If you know the shape of the typical cubic, it starts low left and rising, reverses direction, reverses again, and goes off rising high right.

So subintervals of increase and decrease are determined by the 2 local extrema, which you find by setting the 1st derivative equal to 0:

F(x) = x^3 + 3x² - 24x + 5
F'(x) = 3x² + 6x - 24
3x² + 6x - 24 = 0
x² + 2x - 8 = 0
(x + 4)(x - 2) = 0

x = -4 and x = 2 are the local max and min, that is, at (-4, 85) and (2, -23).So F(x) is increasing on (-∞, -4) and (2, ∞) and decreasing on (-4, 2).

The point of inflection is found by setting the 2nd derivative to 0:

F'(x) = 3x² + 6x - 24
F"(x) = 6x + 6
6x + 6 = 0
6x = -6
x = -1

So the graph is concave down on (-∞, -1) and concave up on (-1, ∞), with the inflection point at (-1, 31).

Answer 2

Is that F(x) or f(x)? The difference is that F(x) is an antiderivative of f(x), or f(x)=F'(x). Either way, the derivative of the function is 3x^2+6x-24. The derivative of that function is 6x+6. So, set that equal to zero, and we get x=-1, so there's a possible point of inflection at x=-1. Test to see what the sign of f"(x) is before and after the critical point of x=-1, and we see that before x=-1, f"(x) is positive, and after x=-1 it's negative. Therefore, there is a point of inflection at x=-1, and the function is concave up before x=-1 and concave down afterwards.

To find the min and max, set f'(x) equal to zero to find critical points of f, then do the same thing with finding the sign of f'(x)' before and after the critical points. If it changes from negative to positive, it's a minimum. If it changes from pos to neg, it's a max.

Answer 3

Use calculus.
F'(x) = 3x^2+6x-24 (the first derivative)
Where F' has zeros, is where F(the original) has local extrema.
The second derivative test tells us whether or not each of the zeros of F' are max or mins of F the original.
if F'(c) = 0
and if (a) F''(c) < 0 , then F has a local maximum at c
if (b) F''(c) > 0 , then F has a local minimum at c

F''(x) = 6x+6 (The second derivative)

Answer 4

Differentiate and set equal to zero to find local max and min.

Differentiate again to find out where the functionis increasing and decreasing.