Question

Statement-1: Let $f(x)$ be a differentiable function such that $f(0)=1, f(1)=2, f(2)=1$ and $f(4)=4$. The graph of $y=f^{\prime}(x)$ is given below.

If $x \in(0,5)$, then the total number of local maximum points and local minimum points of $f(x)$ are two.
Statement-2: For any non constant differentiable function $g(x)$ in $x \in(a, b)$, if $g(x)$ has local maximum at $x=c_1 \in(a, b)$ and local minimum at $x=c_2 \in(a, b)$ then sign of $g^{\prime}(x)$ must change from positive to negative and negative to positive moving from left to right in neighbourhood of $x=c_1$ and $x=c_2$ respectively.

• A. Statement-1 is true, statement-2 is true and statement- 2 is correct explanation for statement- 1 .
• B. Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
• C. Statement-1 is true, statement- 2 is false.
• D. Statement-1 is false, statement-2 is true.

Answer 1

Answer: (A)

Statement-1: $f(x) $ increases from$ x = 0$ to $x = 1$ and $ x = 3$ to $x = 5$. $f(x)$ decreases from $x = 1$ to $x = 3$
So, $x = 1$ is point of local maximum and $x = 3$ is point of local minimum.
Statement-2 is obviously true and also explaining statement-1