Interval question

Question

Determine the intervals on which the given function is concave up or down. Identify the inflection points.

f(x)=x⁴-24x²+1

Answer 1

Graph this first!

Local maximum at 0, local minimum at 2 sqrt(3)

You can find the roots by making the substitution y=x².

Then the concave portions (up and down) are between the roots!

Answer 2

f(x) = x⁴ - 24x² + 1

f'(x) = 4x³ - 48x

f"(x) = 12x² - 48 = 12(x² - 4) = 12(x - 2)(x + 2)

f"(x) = 0 when x = 2 or -2

When x < -2, f"(x) > 0. So, f is concave up

When -2 < x < 2, f"(x) < 0. So, f is concave down

When x . 2, f"(x) > 0. So, f is concave up

f is concave down on (-2,2) and is concave up on (-∞, -2) ∪ (2, ∞)