# How do you find local maximum value of f using the first and second derivative tests: #f(x) = x + sqrt(9 − x) #?

graph{x+sqrt(9-x) [7.9024, 9.588, 8.735, 9.5774]}

By signing up, you agree to our Terms of Service and Privacy Policy

To find the local maximum value of ( f(x) = x + \sqrt{9 - x} ) using the first and second derivative tests:

- Find the derivative of ( f(x) ), ( f'(x) ).

[ f'(x) = 1 - \frac{1}{2\sqrt{9 - x}} ]

- Set ( f'(x) ) equal to zero and solve for critical points.

[ 1 - \frac{1}{2\sqrt{9 - x}} = 0 ]

[ \frac{1}{2\sqrt{9 - x}} = 1 ]

[ \sqrt{9 - x} = 2 ]

[ 9 - x = 4 ]

[ x = 5 ]

- Test the critical point ( x = 5 ) using the second derivative test.

[ f''(x) = \frac{d}{dx}\left(1 - \frac{1}{2\sqrt{9 - x}}\right) ]

[ f''(x) = \frac{1}{4(9 - x)^{\frac{3}{2}}} ]

Since ( f''(5) > 0 ), the function has a local minimum at ( x = 5 ).

Therefore, there is no local maximum value for ( f(x) = x + \sqrt{9 - x} ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided 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 one-sided 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 one-sided 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.

- Is #f(x)=-3x^3+4x^2+3x-4# concave or convex at #x=-1#?
- How do you sketch the graph #y=sqrt(1+x^2)# using the first and second derivatives?
- How do you find the inflection points of the graph of the function: # f(x)=x^(1/3)#?
- How do you find the x coordinates of all points of inflection, final all discontinuities, and find the open intervals of concavity for #y=(2x+3)^2(x+1)^2# for #[-10,0]#?
- How do you sketch the curve #y=x^2+1/x# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7