Question

Given f '(x) = (x + 1)(6 + 3x), find the x-coordinate for the relative minimum on the graph of f(x)

Answer 1

@Nnesha

Answer 2

Basically, set the derivative equal to 0 and solve.

Answer 3

I recommend solving it for zero to give you stationary points.
Then find the second derivative and sub in the stationary pts till you find one that yields f''(x) > 0 (concave up minimum)

Answer 4

Or, you can do some interval testing if you don't wanna find the second derivative.

Answer 5

would the correct answer be 2?

Answer 6

i mean -2

Answer 7

calculate the second derivative first, by expanding the binomial. Then sub in and check.

Answer 8

Another method, is just test your slope near -2 and -1. Reason for this, you know the slope is zero at x = -2 and -1

Answer 9

so how do you determine which one it is

Answer 10

so, your minimum is when x = -1
http://wolframalpha.com/input/?i=y%3D+(x+%2B+1)(6+%2B+3x)

Answer 11

The minimum looks like -1.5 on the graph

Answer 12

ok you have two stationary pts: -1 or -2 right?

f'(x) = (x + 1)(6 + 3x) = x(6+3x) + (6+3x) = 9x + 3x^2 + 6
f''(x) = 9 + 6x
f''(-1) = 9 + 6(-1) = 3 > 0 so concave up --> minimum turning pt
f''(-2) = 9 + 6(-2) = -3 < 0 so concave down --> maximum turning pt

Answer 13

Because when x = -1.1, you have negative slope and when x = 0, you have positive slope. That's an upward U shape, which is a minimum kind of curve.

Answer 14

As for -2, you have a concave down kind of curve, which is a maximum kind of curve.

Answer 15

remember when f'(x) = 0 and a point satisfies this, you have three options
1) minimum turning pt, when f''(x) > 0
OR when f'(x) < 0 on LHS (slope down on left) and f'(x) > 0 on RHS around the pt (slope up on right)
2) maximum turning pt when f''(x) < 0
Or when f'(x) > 0 on LHS and f'(x) < 0 on RHS around the pt.
3) horizontal inflexion when f''(x) = 0 but f''(x) < 0 on LHs and f'''(x) > 0 on RHS or vice versa (so concavity sign changes around the pt)
Or when f'(x) < 0 on LHS and f'(x) < 0 on RHS or f'(x) > 0 on LHS and RHS > 0 (slope direction remains same on LHS and RHS of the pt)