What are all values of x for which the graph of f(x) = (x + 3)(x - 4)^2 is concave downward. = -2/3 < x < 4?

Question

i aslo came up with -3 < x < 4


or is it another solution?

Answer 1

f(x) = (x + 3)(x - 4)^2

To solve for f'(x), use the product rule.

f'(x) = (1)(x - 4)^2 + (x + 3)(2)(x - 4)
f'(x) = (x - 4)^2 + 2(x - 3)(x - 4)

Factoring out (x - 4),

f'(x) = (x - 4) [x - 4 + 2(x - 3)]
f'(x) = (x - 4) [x - 4 + 2x - 6]
f'(x) = (x - 4) (3x - 10)

We require the second derivative to determine concavity. To solve for f''(x), use the product rule.

f''(x) = (1)(3x - 10) + (x - 4)(3)
f''(x) = 3x - 10 + 3(x - 4)
f''(x) = 3x - 10 + 3x - 12
f''(x) = 6x - 22

Setting f''(x) to 0,

0 = 6x - 22
22 = 6x, so
22/6 = x, or
x = 11/3

We determine concavity by testing a value to the right of 11/3, and a value to the left of 11/3. If we get a positive result, it is concave up; a negative result means it is concave down on that interval.

Test x = 0: then f''(0) = 6(0) - 22 = -22, so it follows that
f(x) is concave down on (-infinity, 11/3)

Test x = 1000000. We'll get a positive result for f''(1000000), so f(x) is concave up on (11/3, infinity).

f(x) is concave down on (11/3 to infinity).

Answer 2

f(x) = (x+5)( (x^2) -4x +4) = (x^3) -4(x^2) +4x +5(x^2) -20x +20 = (x^3) +(x^2) -16x +20 f'(x) = 3(x^2) +2x -sixteen equalise this to 0 to acquire the table sure factors 3(x^2) +2x -sixteen = 0 (3x +8) (x -2)= 0 x= -8/3 , +2 to now that's the min. do a 2d differentiation f"(x) = 6x +2 if the fee of this function is effective, while substituting the x with the values gained, then this can provide the x-coordinate of the min. so substituting x=2, f"(x) = 14 so the min. has an x-coordinate of two, on an identical time as the max. has an x-coordinate of -8/3. The graph is concave downward between those 2 values with the aid of fact the curve is shifting from a max. factor to a min. factor. so -8/3