Posted by Ronnie on Monday, November 29, 2010 at 2:26pm.
Briefly, the MVT states that given an arc of a differentiable curve, there is at least one point on that arc at which the slope of the curve is equal to the "average" derivative of the arc, namely the chord between the end-points.
First, see if the function is differentiable between the end-points:
f(x)=√(16-x²) on [-4,1]
f(x) exists and is continuous at all points between -4 and 1. There are no vertical asymptotes or other discontinuities. There is a maximum at x=0.
The slope of the chord
= (f(1)-f(-4))/(1-(-4))
= (√15 - √0)/5
= √15 /5
Differentiate f(x) to get f'(x)
f'(x)= -x/&radic(16-x²)
If the mean-value theorem is valid for this interval, at least one solution exists for f'(x)=√15 / 5
Let's see:
-x/&radic(16-x²) = √15 /5
cross multiply:
-5x = √(15*16-15x²)
Square both sides (and check all roots afterwards):
25x² = 240-15x²
x²=6
x=±√6
since x=+√6 > 1 and is outside of the given interval, it is rejected.
So x=-√6
I will leave it to you to check that the mean-value theorem is satisfied.
Related Questions
calculus - Verify that the hypotheses of the Mean-Value Theorem are satisfied on...
calculus - Verify that the hypotheses of Rolle’s Theorem are satisfied for ...
Calculus - Verify the hypothesis of the mean value theorem for each function ...
calculus - verify that the function satisfies the hypothesis of the mean value ...
math - Verify that f(x) = x^3 − 2x + 6 satisfies the hypothesis of the...
calculus - determine whether the mean value theorem can be applied to f on the ...
math - very urgent ! - Verify that f(x) = x^3 − 2x + 6 satisfies the ...
math - verify that the function satisfies the hypothesis of the mean value ...
calculus - Show that the function f(x)=4x^3−15x^2+9x+8 satisfies the ...
calculus - Find the values of c that satisfy the Mean Value Theorem for f(x)=6/x...
For Further Reading
