Question

Which of the following is true for the function $f\left(x\right)=5x^3-81x^2+11x+97$ over its domain $R$

• A. convex for $x>5.4$
• B. convex for $x<5$
• C. concave for $x\ge 5$
• D. concave for $x\ge 5.4$

Answer 1

The correct option is A convex for $x>5.4$

Obtain the second derivative and find out for what values of $x$ it is positive.
$f^{\prime }\left(x\right)=15x^2-162x+11$
$f^{\prime \prime }\left(x\right)=30x-162$
$f^{\prime \prime }\left(x\right)>0\Rightarrow 30x-162>0\Rightarrow x>5.4$
As $f^{\prime \prime }\left(x\right)>0\forall x>5.4,$ the function is convex for those values, and concave $\forall x<5.4$