Answer to Question #95333 in Real Analysis for Rajni

Question #95333
Identify the intervals in which the function f on R defined by
f(x) = 4x^3-6x^2-72x+15 is both increasing and decreasing ?
1
Expert's answer
2019-10-01T11:18:55-0400

To determine intervals where the function strictly increases or decreases, we need to first find the derivative of the function

"f(x) = 4 x^3 - 6 x^2 - 72 x + 15"


"\\frac{d}{dx}f(x)=\\frac{d}{dx}(4 x^3 - 6 x^2 - 72 x + 15) = 12 x^2 -12x -72=12( x^2 -x -6)=12(x-3)(x+2)" .


Putting "f'(x)=0."

"12(x-3)(x+2)=0 \\\\\n(x-3)(x+2)=0"

So "x=3" and "x=-2" .

The points divide the real line into 3 disjoint intervals, i.e.

"(-\\infty; -2)\\bigcup (-2;3)\\bigcup (3;\\infty)"

"\\def\\arraystretch{1.5}\n \\begin{array}{c|c|c|c}\n \\text{ interval} & (-\\infty;-2) & (-2;3) &(3;\\infty) \\\\ \\hline\n \\text{value } x& x<-2 & -2<x<3 & x>3 \\\\ \\hline\n\\text{sign of } f'(x)=12(x-3)(x+2)& (-)(-)=+>0&(-)(+)=-<0&(+)(+)=+>0\\\\ \\hline\n \\text{nature of function } f & \\text{ increasing } &\\text{ decreasing} & \\text{ increasing }\n\\end{array}"

f(x) is increasing in "(-\\infty; -2)\\bigcup (3;\\infty)"

f(x) is decreasing in "(-2;3)"



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