Let f(x)=5x3−60x+5 input the interval(s) on which f is increasing. (-inf,-2)u(2,inf) input the interval(s) on which f is decreasing. (-2,2) find the intervals on which f is concave up. (-inf,-2)u(2,inf) find the intervals on which f is concave down.

Answers:

(a) f is increasing at .

(b) f is decreasing at .

(c) f is concave up at 

(d) f is concave down at

Explanations:

(a) f is increasing when the derivative is positive. So, we find values of x such that the derivative is positive. Note that



So,



The zeroes of (x - 2)(x + 2) are 2 and -2. So we can obtain sign of (x - 2)(x + 2) by considering the following possible values of x:

-->> x < -2
-->> -2 < x < 2
--->> x > 2

If x < -2, then (x - 2) and (x + 2) are both negative. Thus, (x - 2)(x + 2) > 0.

If -2 < x < 2, then x + 2 is positive but x - 2 is negative. So, (x - 2)(x + 2) < 0.
 If x > 2, then (x - 2) and (x + 2) are both positive. Thus, (x - 2)(x + 2) > 0.

So, (x - 2)(x + 2) is positive when x < -2 or x > 2. Since



Thus, f'(x) > 0 only when x < -2 or x > 2. Hence f is increasing at .

(b) f is decreasing only when the derivative of f is negative. Since



Using the similar computation in (a), 



Based on the computation in (a), (x - 2)(x + 2) < 0 only when -2 < x < 2.

Thus, f'(x) < 0 if and only if -2 < x < 2. Hence f is decreasing at (-2, 2)

(c) f is concave up if and only if the second derivative of f is positive. Note that



Since,



Therefore, f is concave up at .

(d) Note that f is concave down if and only if the second derivative of f is negative. Since,



Using the similar computation in (c), 



Therefore, f is concave down at .