# Determine where the graph of the function is concave upward and where it is concave downward. (Enter your answers using interval notation. If the answer cannot be expressed as an interval, enter EMPTY or ∅.)f(x) = 3x4 − 30x3 + x − 4

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Determine where the graph of the function is concave upward and where it is concave downward. (Enter your answers using interval notation. If the answer cannot be expressed as an interval, enter EMPTY or ∅.)

f(x) = 3x4 − 30x3 + x − 4
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Step 1

Given,

Step 2

The graph of y = f (x) is concave upward on those intervals where its second derivative is greater than zero and the graph of

y = f (x) is concave downward on those intervals where its second derivative is less than zero.

Therefore differentiating given function with respect to x, we get help_outlineImage Transcriptionclosed (x*)nx&(c) = 0, ceR dx f(x)3(4x)-30(3x')+1-0 12x3-90x1 Again differntiating with respect to x, we get f"(x)12(3x)-90(2x)+0 =36x2-180x 36(x-5x) fullscreen
Step 3

Now computing the values of x for which second deriva... help_outlineImage Transcriptionclosef"(x) 0 36(x-5x)> 0 x-5x>0 x(x-5)>0 (x >0 and x-5>0) or (x<0 and x-5<0) (x>0 and x>5) or (x<0 and x <5) xe (5, 00) or xe(-0,0) x(,0)U(5, ) fullscreen

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