Brief calculus.

Question

1. sketch f(x) = x^3-5x-2.6
2. find f(5)
3. find any maximum of f(x)
4. find any minimums of f(x)
5. find the x-intercepts of f(x)
6. find the y-intercept of f(x)
7. find the values of x so that f(x) = -4
8. where is f (x) increasing? where is f (x) decreasing?

Answer 1

Doing 2-8 first will make sketching the graph a lot easier. (Of course, you should use Desmos or any graphing utility to confirm the graph you generate by hand.)

2) Plug in 5 for x in the function equation

6) Plug in 0 for x in the function equation

5) It has 3 real x-intercepts though none of them appear to be rational. Use a graphing utility.

7) Set f(x) = - 4 then add 4 to both sides to get x³ - 5x + 1.4 = 0. Similar to 5) above, 3 solutions all irrational.

All of the above is precalculus. 3, 4, and 8 require the calculus tool of differentiation:

Using power rule, f^'(x) = 3x² - 5. By setting the first derivative = 0 and solving, we can find the local max and the local min, as well as the intervals on which the function increases and decreases. All of these can be confirmed easily with Desmos or another graphing utility.

3x² - 5 = 0. x = ±√(5/3) the negative sq. root of 5/3 is the x-value of the local max, and you can plug it in to f(x) to get the y-value. Likewise, the positive sq. root of 5/3 is the x-value of the min, and f(√(5/3)) is the local minimum. (Note, because f is an odd degree polynomial, its range is all reals and does not have global extrema.) f increases on (-∞ , -√(5/3)) ∪ (√(5/3) , ∞) and is decreasing elsewhere.