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 $-5<x<2$ and $x>4$ 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 $x=-5f^{\prime }(x)$ changes from $+ve$ to $-ve$ 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