How do you determine whether the function f(x)=−6x is concave up or concave down and its intervals?

Step 1In general, to investigate concavity of the graph of function f, we investigate the sign of the second derivative.f(x)=−6x12Note first that the doamin of f is [0, ∞)f′(x)=−3x−12f″(x)=32x−32=32x3f″(x) is positive for all real x, so it is positive for all x in the domain of f.The graph of f is concave up on (0, ∞)(Intervals of concavity are generally given as open intervals.)

Best solutions to - "How do you determine whether the function f(x)=−6x..."

High School
Calculus and Analysis
Calculus 1
Analyzing functions

Ansley Sparks

Answered question

2022-04-16

How do you determine whether the function

f (x) = - 6 \sqrt{x}

is concave up or concave down and its intervals?

Answer & Explanation

ktetik7aeg

Beginner2022-04-17Added 10 answers

Step 1
In general, to investigate concavity of the graph of function f, we investigate the sign of the second derivative.

f (x) = - 6 x^{\frac{1}{2}}

Note first that the doamin of f is

[0, \infty)

f^{'} (x) = - 3 x^{- \frac{1}{2}}

f^{″} (x) = \frac{3}{2} x^{- \frac{3}{2}} = \frac{3}{2 {\sqrt{x}}^{3}}

f^{″} (x)

is positive for all real x, so it is positive for all x in the domain of f.
The graph of f is concave up on

(0, \infty)

(Intervals of concavity are generally given as open intervals.)

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