How do you determine whether the function #f(x)= 6 sqrt (x)# is concave up or concave down and its intervals?
Use calculus (the sign of the second derivative) or algebra/precalculus graphing techniques.
Calculus
(Intervals of concavity are generally given as open intervals.)
Algebra/Precalculus
The graph of the square root function looks like this:
graph{y = sqrtx [10, 10, 5, 5]}
That graph is concave down.
graph{y = 6sqrtx [6.41, 25.63, 13.2, 2.82]}
The graph is concave up.
By signing up, you agree to our Terms of Service and Privacy Policy
To determine the concavity of the function ( f(x) = 6\sqrt{x} ), you need to analyze its second derivative.

Find the first derivative ( f'(x) ) of ( f(x) ) using the power rule and chain rule. ( f'(x) = 6\frac{1}{2\sqrt{x}} = \frac{3}{\sqrt{x}} )

Find the second derivative ( f''(x) ) by differentiating ( f'(x) ) with respect to ( x ). ( f''(x) = \frac{3}{2x^{3/2}} )
Now, determine the sign of ( f''(x) ) to identify concavity:
 If ( f''(x) > 0 ), the function is concave up.
 If ( f''(x) < 0 ), the function is concave down.
For ( f''(x) = \frac{3}{2x^{3/2}} ), it's clear that ( f''(x) > 0 ) for all ( x > 0 ). Thus, the function ( f(x) = 6\sqrt{x} ) is concave up for all positive ( x ) values.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 How do you sketch the curve #y=(x+5)^(1/4)# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
 How do you sketch the graph #y=1/(1+x^4)# using the first and second derivatives?
 Is #f(x)=(x3)(x2)x^2+x^3# concave or convex at #x=1#?
 What is the second derivative of #f(x)=cot^2x #?
 What is the second derivative of #f(x)= ln (x^2+2)#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7