Question

Consider the function q(x)=x^11−3x^10+2. Find the x-coordinates of all local maxima. If there are multiple values, give them separated by commas. If there are no local maxima, enter ∅.

Answer 1

Let x = c is the critical point of f(x) it means f'(x) = 0 at x = c than according to first derivative test we can write,

1) if f'(x) changes from positive to negative at x = c than x = c is local maximum

2) if f'(x) changes from negative to positive at x = c than x = c is local minimum

we have,

we can write,

equate it to 0 we can write,

Hence we can say that x = 0 and x = 30/11 are critical points of the function q(x)

Divide the interval (-infinity, infinity) into three intervals given by (-infinity, 0), (0, 30/11) and (30/11, infinity)

now check the sign of q'(x) on these interval

To check the sign of q'(x) on interval (-infinity, 0) evaluate q'(x) at x = -1 and x = -2

we can write,

Hence we can say that q'(x) > 0 means positive on the interval (-infinity, 0)

To check the sign of q'(x) on interval (0, 30/11) evaluate q'(x) at x = 1 and x = 2

we can write,

Hence we can say that q'(x) < 0 means negative on the interval (0, 30/11)

To check the sign of q'(x) on interval (30/11, infinity) evaluate q'(x) at x = 3 and x = 4

we can write,

Hence we can say that q'(x) > 0 means positive on the interval (30/11, infinity)

we know that local extremum is the value of x where f'(x) changes its sign

we know that to find the x coordinates of local maxima it means according to first derivative test we have to find the value of x at which f'(x) changes from positive to negative

we can write,

q'(x) > 0 means positive on the interval (-infinity, 0)

q'(x) < 0 means negative on the interval (0, 30/11)

q'(x) > 0 means positive on the interval (30/11, infinity)

we can see that at x = 0 f'(x) changes from positive to negative hence we can say that x = 0 is the local maximum

we can write local maximum is given by x = 0

Note : here we have check random values of x to find the sign of f'(x) on respective interval we can take any x value in those intervals.