Statements S and R are as follows:
S : sin x and cosx both are decreasing functions in ,
2
R : if a differentiable function is decreasing in a,b then its derivative is also decreasing in a,b
Which of the following statements is correct?
(1) Both S and R are false
(2) Both S and R are true but R is not correct explanation of S
(3) S is true and R is correct explanation of S
(4) S is true but R is false
The correct answer is (3) S is true and R is correct explanation of S.
S states that sin(x) and cos(x) are both decreasing functions in the interval [0,π/2]. This is true because in this interval, both sin(x) and cos(x) start at their maximum value (1) and decrease as x increases.
R states that if a differentiable function is decreasing in an interval (a,b), then its derivative is also decreasing in that interval. This is true because if a function is decreasing, its derivative will be negative, indicating a decrease in the rate of change.
Therefore, S is true and R is a correct explanation of S.
The correct answer is (4) S is true but R is false.
Statement S is true because sin(x) and cos(x) are both decreasing functions in the interval [0, π/2].
However, statement R is false. Just because a differentiable function is decreasing in the interval (a, b), it does not mean that its derivative is also decreasing in that interval. Counterexamples can be found by considering functions with points of inflection.
To answer this question, we need to analyze both statements S and R.
Statement S claims that both sin x and cos x are decreasing functions in the interval [0, π/2]. To determine if this is true, we can look at the graphs of sin x and cos x in this interval.
The graph of sin x starts at 0 when x = 0 and decreases as x increases until it reaches its minimum value of -1 at x = π/2. Therefore, sin x is indeed a decreasing function in the interval [0, π/2].
The graph of cos x starts at 1 when x = 0 and also decreases as x increases until it reaches its minimum value of 0 at x = π/2. Therefore, cos x is also a decreasing function in the interval [0, π/2].
Since both sin x and cos x are indeed decreasing functions in the interval [0, π/2], statement S is true.
Now let's analyze statement R. It claims that if a differentiable function is decreasing in the interval (a, b), then its derivative is also decreasing in the interval (a, b).
This statement is actually a true statement known as the Corollary to the Mean Value Theorem. According to this corollary, if a function f(x) is differentiable on an interval (a, b) and f'(x) < 0 for all x in (a, b), then f(x) is decreasing on the interval (a, b).
Since statement R is a correct explanation of statement S, we can conclude that option (3) "S is true and R is the correct explanation of S" is the correct answer.