Question

For what value of a ,f(x)= -x^3+4ax^2+2x-5 is decreasing function

Answer 1

f(x) = -x³ + 4ax² + 2x - 5 is decreasing function .
it means f'(x) < 0

now, differentiate f(x) with respect to x
$f'(x)=-3x^{3-1}+4\times\:2ax^{2-1}+2x^{1-1}-0\\f'(x)=-3x^2+8ax+2 < 0 \\ 3 {x}^{2} - 8ax - 2 > 0$
this is possible only when,
coefficient of x² > 0 and Discriminant < 0
here, coefficient of x² = 3 > 0
and D = (-8a)² -4(-2)(3) < 0
64a² + 24 < 0 it can't possibly for real value of a .

hence, a ∈ Φ