Posted by Anonymous on Friday, November 9, 2012 at 10:51am.
It is given that a differentiable function f(x)=4x^3+kx^236x15 (k is a constant) is decreasing on 3/2<x<2 and increasing on 2<x<5. Find the value of k and the turning point(s) of the curve y=f(x).

Calculus  Steve, Friday, November 9, 2012 at 11:28am
you know that
f'(x) has a root at x=2.
It is negative for 3/2 < x < 2
It is positive for 2<x<5
Nothing is said about any domain outside (3/2,5)
But, since you want a cubic f(x), f'(x) must be positive for x < 3/2, so it has another root at x = 3/2
f'(x) = a(2x+3)(x2) = a(2x^2  x  6)
now we can get somewhere
f(x) = a(2/3 x^3  1/2 x^2  6x) + c
or, changing a and c (which are arbitrary) and clearing fractions,
f(x) = a(4x^3  3x^2  36x) + c
Now, we want f(x) as described above, so a = 1 and
f(x) = 4x^3  3x^2  36x  15