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Calculus

Differentiability

Concept Quizzes

Approximate Rate of Change
Slope of a Curve
Tangent Line at a Point
Local Linear Approximation
Intermediate Value Theorem
Mean Value Theorem
Rolle's Theorem
Characteristics of graphs of f and f'

Challenge Quizzes

Differentiability: Level 2 Challenges
Differentiability: Level 3 Challenges
Differentiability: Level 4 Challenges

Characteristics of graphs of f and f'

Given $f'(x) = x^2 + 9x + k,$ if $f(x)$ is an increasing function for all values of $x,$ what is the range of $k?$

\displaystyle{\frac{81}{4} \leq k}

\displaystyle{\frac{77}{4} \leq k}

\displaystyle{\frac{83}{4} \leq k}

\displaystyle{\frac{79}{4} \leq k}

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by Brilliant Staff

Suppose $f(x)$ is a function such that $y=f'(x)$ is given by the curve in the above diagram. If $y= f'(x)$ is tangent to the $x$ -axis at $x=a=7,$ which of the following statements is NOT correct about $f(x)?$

There is at most one point of intersection with the

x

-axis. f(x) is decreasing for all values of

x.

f(7)

is a local maximum. If

f(7)

0,

f(0)

is positive.

Show explanation

by Brilliant Staff

Suppose $f(x)$ is a differentiable function such that the graph of $y=f'(x)$ is as shown in the above diagram. If $a = 2, b = 3, c = 5, d= 7,$ and $f'(e) = f'(d)$ for all $e \geq d,$ which of following statements is correct?

[-2, 2],

f(x)

attains a maximum value at

f(0)

f(5)

is a local maximum. There are four extreme values between

x=-3

and

x=7.

f(x)

is a linear function for

x > 7 .

Show explanation

by Brilliant Staff

Suppose $f(x)$ is a function such that the graph of $y=f'(x)$ is a line as shown in the above diagram. If $a=19,$ which of following statements is correct?

f(x)

is an increasing function for

x \le 19

f(19)

is a local maximum.

f(x)

is an increasing function for

x \ge 19

f(19)=0.

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by Brilliant Staff

Suppose $f(x)$ is a function such that $f'(x) = K(x-a)(x-b) ,$ where $K$ is a positive constant. If $a=5$ and $b=10,$ which of following statements is NOT correct?

f(x)

is a cubic function. If

f(5)= 0,

there are two intersection points between

f(x)

and the

x

-axis.

f(5)

and

f(10)

are always

0.

f(x)

is a decreasing function on

[5,10]

Show explanation

by Brilliant Staff