Question

Which of the following is (are) true about the function $f\left(x\right)=-\frac{3}{4}{x}^4-8x^3-\frac{45}{2}{x}^2+105$ ?

• A. x = 0 is point of local maxima
• B. x = 0 is point of local minima
• C. x = –3 is point of local minima
• D. x = –5 is point of local maxima.
• E. x = –5 is point of local minima.

Answer 1

Answer: (A), (C), (D)

x = –5 is point of local maxima.

$f\text{'}\left(x\right)=-3x^3-24x^2-45x=-3x(x^2+8x+15)\Rightarrow f\text{'}\left(x\right)=-3x\left(x+3\right)\left(x+5\right)\Rightarrow f\text{'}\left(x\right)=0\Rightarrow x=0,-3,-5f\text{'}\text{'}\left(x\right)=-9x^2-48x-45=-3\left(3x^2+16x+15\right)f\text{'}\text{'}\left(0\right)=-45<0.\mathrm{Therefore},x=0\mathrm{is}\mathrm{point}\mathrm{of}\mathrm{local}\mathrm{maxima}f\text{'}\text{'}\left(-3\right)=18>0\mathrm{Therefore},x=-3\mathrm{is}\mathrm{point}\mathrm{of}\mathrm{local}\mathrm{minima}f\text{'}\text{'}\left(-5\right)=-30<0\mathrm{Therefore},x=-5\mathrm{is}\mathrm{point}\mathrm{of}\mathrm{local}\mathrm{maxima}.$