Question

Let $f$ be differential for all $x$. If $f\left(1\right)=-2$ and $f^{\prime }\left(x\right)\ge 2$ for $xϵ[1,6]$, then ?

• A. $f\left(6\right)=5$
• B. $f\left(6\right)<5$
• C. $f\left(6\right)<8$
• D. $f\left(6\right)\ge 5$

Answer 1

The correct option is A $f\left(6\right)=5$

As f(1)=−2f(1)=−2 and f′(x)≥2∀x∈[1,6]f′(x)≥2∀x∈[1,6]

Applying Langrange's mean value theorem

f(6)−f(1)5f(6)−f(1)5=f′(c)≥2=f′(c)≥2

⇒f(6)≥10+f(1)⇒f(6)≥10+f(1)

⇒f(6)≥10−2⇒f(6)≥10−2

⇒f(6)≥8