Question

The diagram shows the graph of the derivative of a function $f(x)$ for $0\le x\le 4$ with $f(0)=0$. Which of the following could be correct statements for $y=f(x)$ ?

• A. Tangent line to $y=f(x)$ at $x=0$ makes an angle of ${\sec }^{-1}\sqrt{5}$ with the $x$-axis.
• B. f is strictly increasing in $(0,3)$.
• C. $x=1$ is both an inflection point as well as point of local extremum.
• D. Number of critical point on $y=f(x)$ is two.

Answer 1

Answer: (A), (B), (D)

Slope of $y$ tangent at $x=0$ is ${\tan }^{-1}(2)\Rightarrow (A)$ is correct
$f$ is obviously increasing for $(0,3)$
$\left(f^{\prime }(x)\ge 0\right)\Rightarrow$(B) is correct
$x=1$ being an inflection point
$x=1$ is inflection but not as extremum as $f^{\prime }(x)$ does not change sign hence (B) is not correct