Question

Let $f \left(1\right)=-2$ and $f' \left(x\right) \ge4.2 for 1\le x\le6.$ The possible value of $f (6)$ lies in the interval :

• A. [15,19)
• B. $\left(-\infty, 12\right)$
• C. [12,15)
• D. [$19, \infty$)

Answer 1

Answer: (D)

Given $f (1) = - 2$ and $f' \left(x\right) \ge 4.2 \, for \, 1\le x\le6$
Consider $f' \left(x\right) =\frac{f \left(x+h\right)-f\left(x\right)}{h}$
$\Rightarrow f\left(x+h\right)-f\left(x\right)=f' \left(x\right). h \ge\left(4.2\right)h$
$So, \, f \left(x+h\right)\ge f\left(x\right)+\left(4.2\right)h$
put x= 1 and h = 5, we get
$f\left(6\right)\ge f\left(1\right)+5\left(4.2\right)\, \Rightarrow \quad f\left(6\right)\ge19$
Hence $ f\left(6\right) lies \, in [19, \infty)$