Please solve fast please
1 answer:
Answer:
see explanation
Step-by-step explanation:
Using identities
sec²x = tan²x + 1
cosec²x = cot²x + 1
cos²x =
(1)
sin²θ +
= sin²θ + cos²θ = 1
(2)
9sec²θ - 9tan²θ
= 9sec²θ - 9(sec²θ - 1)
= 9sec²θ - 9sec²θ + 9
= 9
(3)
- 8cot²A + 8cosec²A
= - 8cot²A + 8(cot²A + 1)
= - 8cot²A + 8cot²A + 8
= 8
(4)
- 10cosec²β + 10cot²β
= - 10(cot²β + 1) + 10cot²β
= - 10cot²β - 10 + 10cot²β
= - 10
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Note that f(x) as given is <em>not</em> invertible. By definition of inverse function,
which is a cubic polynomial in with three distinct roots, so we could have three possible inverses, each valid over a subset of the domain of f(x).
Choose one of these inverses by restricting the domain of f(x) accordingly. Since a polynomial is monotonic between its extrema, we can determine where f(x) has its critical/turning points, then split the real line at these points.
f'(x) = 3x² - 1 = 0 ⇒ x = ±1/√3
So, we have three subsets over which f(x) can be considered invertible.
• (-∞, -1/√3)
• (-1/√3, 1/√3)
• (1/√3, ∞)
By the inverse function theorem,
where f(a) = b.
Solve f(x) = 2 for x :
x³ - x + 2 = 2
x³ - x = 0
x (x² - 1) = 0
x (x - 1) (x + 1) = 0
x = 0 or x = 1 or x = -1
Then can be one of
• 1/f'(-1) = 1/2, if we restrict to (-∞, -1/√3);
• 1/f'(0) = -1, if we restrict to (-1/√3, 1/√3); or
• 1/f'(1) = 1/2, if we restrict to (1/√3, ∞)