Consider the equation . how many different solutions does this equation have if all the variables must be positive integers? ent
1 answer:
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Answer:
1) increasing on (-∞,-1] ∪ [1,∞), decreasing on [-1,0) ∪ (0,1]
is local maximum,
is local minimum
2) increasing on [1,∞), decreasing on (-∞,0) ∪ (0,1]
is absolute minimum
3) increasing on (-∞,0] ∪ [8,∞), decreasing on [0,4) ∪ (4,8]
is local maximum,
is local minimum
4) increasing on [2,∞), decreasing on (-∞,2]
is absolute minimum
5) increasing on the interval (0,4/9], decreasing on the interval [4/9,∞)
is local minimum,
is absolute maximum
Step-by-step explanation:
To find minima and maxima the of the function, we must take the derivative and equalize it to zero to find the roots.
1)
and
So, the roots are and
The function is increasing on the interval (-∞,-1] ∪ [1,∞)
The function is decreasing on the interval [-1,0) ∪ (0,1]
is local maximum,
is local minimum.
2)
and
So the root is
The function is increasing on the interval [1,∞)
The function is decreasing on the interval (-∞,0) ∪ (0,1]
is absolute minimum.
3)
and
So the roots are and
The function is increasing on the interval (-∞,0] ∪ [8,∞)
The function is decreasing on the interval [0,4) ∪ (4,8]
is local maximum,
is local minimum.
4)
has no solution and
is crtitical point.
The function is increasing on the interval [2,∞)
The function is decreasing on the interval (-∞,2]
is absolute minimum.
5) for
So the root is
The function is increasing on the interval (0,4/9]
The function is decreasing on the interval [4/9,∞)
is local minimum,
is absolute maximum.