Suppose f(x)= |x^(2)+2x−63|.
Where is f(x) not differentiable?
f(x) = |(x+9)(x-7)|
the function y = (x+9)(x-7) would have x-intercepts at -9 and 7 , and it would dip below the x-axis between these two values.
So the |...| flips that part above the x-axis, causing CUSPS at x =-9 and x=7
the derivative does not exist for those two values.
Hmm, DNE was incorrect.
To determine where the function f(x) is not differentiable, we need to find the points where the function does not satisfy the conditions required for differentiability.
The function f(x) = |x^2 + 2x - 63| is composed of two separate functions based on the sign of the expression inside the absolute value.
When x^2 + 2x - 63 is positive (x^2 + 2x - 63 > 0), f(x) simplifies to f(x) = x^2 + 2x - 63.
When x^2 + 2x - 63 is negative (x^2 + 2x - 63 < 0), f(x) simplifies to f(x) = -(x^2 + 2x - 63).
Now, let's find the points where f(x) may not be differentiable.
1. First, find the critical points by setting the expression inside the absolute value to zero:
x^2 + 2x - 63 = 0
Factorizing the quadratic equation, we have:
(x - 7)(x + 9) = 0
This gives two critical points:
x = 7 and x = -9.
2. Determine the intervals based on the critical points:
For x < -9, x^2 + 2x - 63 < 0, which means f(x) = -(x^2 + 2x - 63).
For -9 < x < 7, x^2 + 2x - 63 > 0, which means f(x) = x^2 + 2x - 63.
For x > 7, x^2 + 2x - 63 > 0, which means f(x) = x^2 + 2x - 63.
Now, let's analyze each interval to check for differentiability:
1. For x < -9:
Since f(x) = -(x^2 + 2x - 63), it is a differentiable function on this interval.
2. For -9 < x < 7:
On this interval, f(x) = x^2 + 2x - 63, which is a quadratic function. Quadratic functions are smooth curves, and therefore, f(x) is differentiable on this interval.
3. For x > 7:
Similar to the second interval, f(x) = x^2 + 2x - 63, which is a quadratic function. Thus, f(x) is differentiable on this interval as well.
Therefore, f(x) is differentiable for all real numbers, and there are no points where f(x) is not differentiable.
To determine where a function is not differentiable, we need to identify the points where the function is either not defined or exhibits sharp turns, cusps, or corners. In the case of the function f(x) = |x^2 + 2x - 63|, we can proceed by following the steps below:
1. First, let's find the critical points of the function by setting the expression inside the absolute value equal to zero and solving for x.
x^2 + 2x - 63 = 0
Factoring the quadratic expression, we have:
(x - 7)(x + 9) = 0
This gives us two critical points: x = 7 and x = -9.
2. Next, let's examine the behavior of the function in the intervals between and outside the critical points.
a) For x < -9, the expression inside the absolute value, x^2 + 2x - 63, is negative. Therefore, f(x) = -(x^2 + 2x - 63) in this interval.
b) Between -9 < x < 7, the expression inside the absolute value, x^2 + 2x - 63, is positive. Hence, f(x) = x^2 + 2x - 63 in this interval.
c) For x > 7, the expression inside the absolute value, x^2 + 2x - 63, is positive as well. Again, f(x) = x^2 + 2x - 63 in this interval.
3. Now, let's analyze the differentiability at each critical point and where the function changes behavior (at -9 and 7).
a) At x = -9, f(x) transitions from negative to positive. This is a point of non-differentiability since the function exhibits a sharp turn at this point.
b) At x = 7, f(x) changes behavior from positive to negative. This is another point of non-differentiability due to a sharp turn.
Therefore, the function f(x) = |x^2 + 2x - 63| is not differentiable at x = -9 and x = 7.