# calculus

posted by .

How many integer values of a are there such that
f(x)=x^+ax^2+8ax+25
has no local extrema?

• calculus -

Assuming you meant

x^3+ax^2+8ax+25

we want a derivative with no zeros. So,

f' = 3x^2 + 2ax + 8a
f" = 6x+2a

this will have no zeros if the discriminant is negative, so we need

(2a)^2 - 4(3)(8a) < 0
4a^2 - 96a < 0
4a(a-24) < 0

So 0<a<24

At a=0 and a=24, f"(0) when f'=0, so there's an inflection point, so no extrema.

So, 0 <= a <= 24

## Similar Questions

1. ### Calculus; Limits

Evaluate limit, x -> a, [(x + 4a)^2 - 25a^2] / [x - a] My work: = limit, x -> a, (x^2 + 8ax + 16a^2 - 25a^2) / (x - a) = limit, x -> a, (x^2 + 8ax - 9a^2) / (x - a) = limit, x -> a, (x + 8a - 9a^2) / (-a) = (a + 8a - 9a^2) …
2. ### Calculus

For any constant c, define the function f_c(x)= x^3+2x^2+cx. (a) Graph y = f_c(x) for these values of the parameter c: c = -1, 0, 1, 2, 3, 4. What are the similarities and differences among the graphs, and how do the graphs change …
3. ### Calculus

For any constant c, define the function f_c(x)= x^3+2x^2+cx. (a) Graph y = f_c(x) for these values of the parameter c: c = -1, 0, 1, 2, 3, 4. What are the similarities and differences among the graphs, and how do the graphs change …
4. ### Calculus

let f(x)=x^3(e^-x) Answer using calculus, use graphing calculator only to check work. a) Find the local and global extrema of f. b) Find the intervals where f is increasing/decreasing. c) Find the inflection points of f. d) Find the …
5. ### math

Given the function f(x)=x^4−6x^2 a.) Find all values of x where local extrema points occur b.)find all values of x where inflection points occur c.) find all local extrema of f(x) d.) find all global extrema of f(x) over the …
6. ### Math

How many x-intercepts and how many local extrema does the polynomial P(x)= x(sqrt)3-4x have?
7. ### Calculus

find the intervals on which f(x) is increasing and decreasing along with the local extrema. f(x)=x^4 + 18x^2 I took the derivative and got: f'(x)= 4x^3 + 36x When I set this to zero, I get the imaginary number 3i. I can't get test …