Question

Suppose you do not know the function $f ( x )$, however some information about $f( x )$ is listed below. Read the following carefully before attempting the questions
(i) $ f( x )$ is continuous and defined for all real numbers
(ii) $ f^{\prime}(-5)=0 ; f^{\prime}(2)$ is not defined and $f^{\prime}(4)=0$
(iii) $(-5,12)$ is a point which lies on the graph of $f ( x )$
(iv) $f^{\prime \prime}(2)$ is undefined, but $f^{\prime \prime}( x )$ is negative everywhere else.
(v) the signs of $f^{\prime}( x )$ is given below

From the possible graph of $y =f( x )$, we can say that

• A. There is exactly one point of inflection on the curve.
• B. $f(x)$ increases on $-54$ and decreases on $-\infty< x< -5$ and $2< x< 4$.
• C. The curve is always concave down.
• D. Curve always concave up.

Answer 1

Answer: (C)

(ii) At $x=-5 f^{\prime}(x)$ changes from $+v e$ to $-v e$ and $x=4, f^{\prime}(x)$ change sign for + ve to - ve hence maxima at $x=-5$ and 4 . $f$ is continuous and $f^{\prime}(x)$ is not defined hence $x=2$ must be geometrical sharp corner