Question

Which of the following is (are) true about the function $f\left(x\right)=4x^3–18x^2+27x–7$ ?

• A. Has a maxima at x = 3/2
• B. x = 3/2 is a point of inflexion
• C. Has a minima at x = 3/2
• D. Neither has a maxima nor minima

Answer 1

Answer: (B), (D)

The correct options are
B x = 3/2 is a point of inflexion
D Neither has a maxima nor minima

$f\left(x\right)=4x^3–18x^2+27x–7\Rightarrow f\text{'}\left(x\right)=12x^2–36x+27\Rightarrow f\text{'}\left(x\right)=3(4x^2–12x+9)=3(2x–3{)}^{2}f\text{'}(x)=0\Rightarrow x=\frac{3}{2}(\mathrm{critical}\mathrm{point})\mathrm{Since}f\text{'}(x)>0\mathrm{for}\mathrm{all}x<\frac{3}{2}\mathrm{and}\mathrm{for}\mathrm{all}x>\frac{3}{2}\mathrm{Hence}x=\frac{3}{2}\mathrm{is}a\mathrm{point}\mathrm{of}\mathrm{inflexion}i.e.,\mathrm{neither}a\mathrm{point}\mathrm{of}\mathrm{maxima}\mathrm{nor}a\mathrm{point}\mathrm{of}\mathrm{minima}.x=\frac{3}{2}\mathrm{is}\mathrm{the}\mathrm{only}\mathrm{critical}\mathrm{point},\mathrm{and}f\mathrm{has}\mathrm{neither}\mathrm{maxima}\mathrm{nor}\mathrm{minima}.$