Question

F:R—>R, f(x)=x⁶+192x+10,Determine intervals in which the given function are strictly increasing or strictly decreasing.

Answer 1

f :R –––> R ,f(x) = x^6 + 192x + 10

f(x) = x^6 + 192x + 10

differentiate with respect to x,

f'(x) = 6x^5 + 192

now, f'(x) = 0

6x^5 = -192 => x^5 = -32

x = -2

case 1 :- x > -2 , f'(x) < 0
so, function is strictly decreasing in (-2, ∞)

case 2 :- x < -2 , f'(x) > 0
so, function is strictly increasing in (-∞ , -2)

hence, function is strictly increasing in (-∞, -2) while function is strictly decreasing in (-2, ∞)

Answer 2

Dear student:

Given: F:R—>R,

f(x)=x⁶+192x+10

For determining the intervals in which f is increasing and decreasing.

Find derivative of f(x)

Then see the derivative in which f is positive and negative.

If it is positive then f is increasing

And if it is negative then f is decreasing.

See the attachment.