Question

Find the range in which the function f(x) = 8 + 40x^3 – 5x^4 – 4x^5 is increasing.

(a) 2<z<0, 0<z<3

(b) 1<z<0, 0<z<2

(c) 3<z<0, 0<z<2

(d) 3<z<0, 0<z<4

I got this question by my college director while I was bunking the class.

I'm obligated to ask this question of Total Derivative in section Partial Differentiation of Engineering Mathematics

Answer 1

Right option is (c) 3<z<0, 0<z<2

The explanation: Given: f(x)=8 + 40x^3 – 5x^4 – 4x^5

f'(x) = 120x^2 – 20x^3 – 20x^4

f'(x) = -20x^2 (x+x^2 – 6)

f'(x) = -20x^2 (x+3)(x-2)

Next, we need to know where the function is not changing and so all we need to do is set the derivative equal to zero and solve.

f'(x) = -20x^2 (x+3)(x-2)=0

From this it is pretty easy to see that the derivative will be zero, and hence the function will not be moving, at,

x=0,-3,-2

Because the derivative is continuous, we know that the only place it can change sign is where the derivative is zero. So, a quick number line will give us the sign of the derivative for the various intervals.

From this we get,

Increasing: 3<z<0, 0<z<2