Question

Suppose the derivative of a function f is f′(x)=(x−5)^5(x−4)^4(x+12)^8. Then the function f is increasing on the interval : ?

Answer 1

when f is increasing, its derivative is positive. so you just need to find the values for x for which f'(x) > 0

Answer 2

Okay I got 5 and 4... is that right?

Answer 3

let me check - one sec...

Answer 4

ok, so we want:\[(x-5)^6(x-4)^4(x+12)^8>0\]the even powers will always produce either a zero or just positive numbers. zero is not allowed, so x=4 and x=-12 should NOT be in the interval.
for \((x-5)^5\), this can be positive, zero or negative, so the only valid values here must be x>5.
we are therefore left with:\[x\neq4\]\[x\neq-12\]\[x>5\]
so the only interval that satifies all three is x>5

Answer 5

\((x-5)^6\) at the top should be \((x-5)^5\)

Answer 6

Can you put this in interval notation?

Answer 7

I believe \(x>5\) can be expressed as \((5,\infty]\)

Answer 8

thanks! ^-^

Answer 9

yw