If a continuous function f(x) does not have a root in the interval [a, b], then which one of the following statements is TRUE?a)f(a). f(b) = 0b)f(a). f(b) 0d)f(a) / f(b) ≤ 0Correct answer is option 'C'. Can you explain this answer?

Question

If a continuous function f(x) does not have a root in the interval [a, b], then which one of the following statements is TRUE?a)f(a). f(b) = 0b)f(a). f(b) < 0c)f(a). f(b) > 0d)f(a) / f(b) &le; 0Correct answer is option 'C'. Can you explain this answer?

Answer

We know that, (Intermediate value theorem)

If f(a)f(b) < 0 then f(x) has at least one root in (a, b)

Since there is no root in [a, b] this implies f(a) and f(b) are of same sign

i.e. either they both are positive or they both are negative

In both cases f(a)f(b) > 0

Answer

If a continuous of the present time OOO f (x)
does (a,b,c ) true is congratulations for us
f (a) . f(b) put o

Answer

Understanding the Problem
In the context of continuous functions, the Intermediate Value Theorem states that if a function is continuous on an interval [a, b] and takes values of opposite signs at the endpoints, then there exists at least one root in that interval.
Analyzing the Given Options
- Option A: f(a) * f(b) = 0
- This implies that either f(a) = 0 or f(b) = 0, indicating that there is a root at one of the endpoints. Thus, this option cannot be true.
- Option B: f(a) * f(b) < />
- This suggests that f(a) and f(b) have opposite signs, which would mean a root exists in the interval [a, b]. Hence, this option is also false.
- Option C: f(a) * f(b) > 0
- This indicates that both f(a) and f(b) are either positive or negative. If f(a) and f(b) have the same sign, and there is no root in [a, b], it confirms that the function does not cross the x-axis within the interval.
- Option D: f(a) / f(b) ≤ 0
- This option implies that one of the function values is non-positive while the other is non-negative, contradicting the assumption that both are of the same sign.
Conclusion
The correct answer is Option C: f(a) * f(b) > 0. This statement holds true when a continuous function does not have a root in the interval [a, b], as it signifies that the function maintains a consistent sign throughout the interval.

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