Question

Points of inflexion of the curve
$y=x^4-6x^3+12x^2+5x+7$ are

• A. $(1,19);(1,12)$
• B. $(1,19);(2,33)$
• C. $(1,2);(2,1)$
• D. $(1,7);(2,6)$

Answer 1

The correct option is B $(1,19);(2,33)$

$y=x^4-6x^3+12x^2+5x+7$

$y\left(x\right)=4x^3-18x^2+24x+5$

$y\left(x\right)=12x^2-36x+24$

$y\left(x\right)=0$

$12x^2-36x+24=0$

$x^2-3x+2=0$

$x^2-2x-x+2=0$

$x(x-2)-1(x-2)=0$

$(x-1)-1(x-2)=0$

$(x-1)(x-2)=0$

$x=1,2$

Inflection point of a function is where the function changes from concave up to concave down or vice-versa

$x<1,f\left(x\right)>0$

$1<x<2,f\left(x\right)<0$

$x>2,f\left(x\right)>0$

$\because f\left(x\right)$ changes sign

$\therefore$ At $x=1,2y=f\left(x\right)$ has inflection point

At $x=1,y=f\left(1\right)=19$

At $x=2,y=f\left(2\right)=33$

Point of inflection $(1,19);(2,33)$