question 7 (1 point)\nthe mean value theorem states that if a function f(x) is continuous on a,b and…

question 7 (1 point)\nthe mean value theorem states that if a function f(x) is continuous on a,b and differentiable on (a,b), then there exists a point c in (a,b) such that the slope of the tangent line at c equals:\nzero\nthe slope of the secant line between a and b\nthe average rate of change of f between 0 and c\nf(c)

question 7 (1 point)\nthe mean value theorem states that if a function f(x) is continuous on a,b and differentiable on (a,b), then there exists a point c in (a,b) such that the slope of the tangent line at c equals:\nzero\nthe slope of the secant line between a and b\nthe average rate of change of f between 0 and c\nf(c)

Answer

Brief Explanations:

The Mean Value Theorem (MVT) formula is (f^{\prime}(c)=\frac{f(b)-f(a)}{b - a}). The right - hand side (\frac{f(b)-f(a)}{b - a}) is the slope of the secant line connecting the points ((a,f(a))) and ((b,f(b))).

  • The slope of the tangent line at (c) is (f^{\prime}(c)).
  • The slope of the secant line between (a) and (b) is (\frac{f(b)-f(a)}{b - a}), which is equal to (f^{\prime}(c)) by the MVT.
  • The slope of the tangent line at (c) is not necessarily zero (that would be Rolle's Theorem when (f(a)=f(b))).
  • The average rate of change between (0) and (c) is (\frac{f(c)-f(0)}{c-0}), which is not related to the MVT formula.
  • (f(c)) is the value of the function at (c), not a slope.

Answer:

the slope of the secant line between a and b