Answer:
a) even
b) 4th differences: -72
c) minimum: 0, maximum: 4. This function has 0 real zeros.
d) -1377
e) -288
Explanation:
You want to know a number of the characteristics of the function f(x) = -3x⁴ +6x² -10:
- whether even or odd
- which finite differences are constant
- number of zeros
- AROC on [2, 7]
- IROC at x=3
a) Even/Odd
A function is even if f(x) = f(-x). The graph of an even function is symmetrical about the y-axis. An even polynomial function will only have terms of even degree.
The exponents of the terms of f(x) are 4, 2, 0. These are all even, so we can conclude the function is an even function.
We can also evaluate f(-x):
f(-x) = -3(-x)⁴ +6(-x)² -10 = -3x⁴ +6x² -10 ≡ f(x) . . . . . the function is even
b) Finite differences
We can look at values of x on either side of x=0. The attachment shows function values and finite differences for x = -3, -2, ..., +3.
The fourth finite differences are constant at -72. (We expect this value to be -3·4!, the leading coefficient times the degree of the polynomial, factorial.)
c) Number of zeros
A 4th-degree polynomial will always have exactly four zeros. They may be complex, rather than real. Complex zeros will come in conjugate pairs, so the number of real zeros may be 0, 2, or 4; a minimum of 0 and a maximum of 4.
This polynomial function has no real zeros. The four complex zeros are approximately ...
±1.18864247 ±0.64255033i
d) AROC on [2, 7]
The average rate of change on the interval [a, b] is given by ...
AROC = (f(b) -f(a))/(b -a)
For [a, b] = [2, 7], this is ...
AROC = (((-3(7²) +6)7² -10) -((-3(2²) +6)2² -10)/(7 -2)
= ((-147 +6)(49) -(-12 +6)(4)) / 5 = (-6909 +24)/5 = -6885/5 = -1377
The average rate of change on [2, 7] is = -1377.
e) IROC at x=3
The derivative of the function is ...
f'(x) = -3(4x³) +6(2x) = 12x(-x² +1)
f'(3) = 12·3(-3² +1) = 36(-8) = -288
The instantaneous rate of change at x=3 is -288.